misc
system:sage

{{{id=15|
K.<a> = NumberField(x^2 + 2)
}}}

{{{id=16|
E = EllipticCurve([a, 3]); E
///
Elliptic Curve defined by y^2  = x^3 + a*x + 3 over Number Field in a with defining polynomial x^2 + 2
}}}

{{{id=17|
E.j_invariant()
///
-3359232/59177*a + 221184/59177
}}}

{{{id=18|
E.simon_two_descent()
///
(0, -1, [])
}}}

{{{id=19|
E.discriminant()
///
128*a - 3888
}}}

{{{id=20|
E.omega
}}}

{{{id=0|
E = EllipticCurve([1,2,3,4,5]); E
///
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
}}}

{{{id=9|
E.cremona_label()
///
'10351a1'
}}}

{{{id=2|
EllipticCurve([1,2])
///
Elliptic Curve defined by y^2  = x^3 + x + 2 over Rational Field
}}}

{{{id=3|
E = EllipticCurve(1); E
///
Elliptic Curve defined by y^2 + x*y  = x^3 + 36/1727*x + 1/1727 over Rational Field
}}}

{{{id=4|
E.j_invariant()
///
1
}}}

{{{id=5|
EllipticCurve('389a1')
///
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
}}}

{{{id=6|
E = EllipticCurve('11a2'); E
///
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field
}}}

{{{id=8|
P = SteinWatkinsPrimeData(0)
}}}

{{{id=10|
C = P.next()
}}}

{{{id=11|
C.curves
///
[[[1, -1, 1, -1, 0], '[1]', '1', '4'], [[1, -1, 1, -6, -4], '[2]', '1', '2x'], [[1, -1, 1, -1, -14], '(4)', '1', '4'], [[1, -1, 1, -91, -310], '[1]', '1', '2']]
}}}

{{{id=12|
cremona_optimal_curves([25..30])   
///
<generator object at 0x2ad8cb0>
}}}

{{{id=13|
list(cremona_optimal_curves([1..30]))
///
[Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 10*x - 10 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - x - 14 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 + x^2 - 9*x - 15 over Rational Field, Elliptic Curve defined by y^2  = x^3 + x^2 + 4*x + 4 over Rational Field, Elliptic Curve defined by y^2 + x*y  = x^3 - 4*x -1 over Rational Field, Elliptic Curve defined by y^2  = x^3 - x^2 - 4*x + 4 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 - 5*x - 8 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 3*x + 3 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + x + 2 over Rational Field]
}}}

{{{id=14|
E = EllipticCurve([1..5])
}}}

{{{id=21|
factor(E.discriminant())
///
-1 * 11 * 941
}}}

{{{id=22|
factor(E.conductor())
///
11 * 941
}}}

{{{id=23|
E.division_polynomial(3)
///
3*x^4 + 9*x^3 + 33*x^2 + 87*x + 35
}}}

{{{id=24|
E.period_lattice()
///
(2.78074001376672977106319762718135849939056146053485715119329728332301647731874484861561725275617768153884906260176267799390732160980543532425139655234517, -1.39037000688336488553159881359067924969528073026742857559664864166150823865937242430780862637808884076942453130088133899695366080490271766212569827617259 + 1.06874977635619306615926354743750387875911499084979920494705102192645898476224667886155180674198541474971969884036919522243630126388318193197261198198844*I)
}}}

{{{id=25|
E.omega()
///
2.78074001376672977106319762718135849939056146053485715119329728332301647731874484861561725275617768153884906260176267799390732160980543532425139655234517
}}}

{{{id=26|
E.root_number()
///
-1
}}}

{{{id=27|
E = EllipticCurve('37b')
}}}

{{{id=28|
E.Lseries_at1(k=10)  # 10 terms
///
(0.725676956622683, 0.0000360967566544175)
}}}

{{{id=31|
E.Lseries_at1(k=100)  # 100 terms
///
(0.725681061936153, 1.52437502288743e-45)
}}}

{{{id=32|
E = EllipticCurve('37a')
}}}

{{{id=33|
E.Lseries_deriv_at1(k=10)  # 10 terms
///
(0.306000959182700, 0.0000360967566544175)
}}}

{{{id=34|
E.Lseries_deriv_at1(k=100)  # 100 terms
///
(0.305999773834879, 1.52437502288740e-45)
}}}

{{{id=35|
E = EllipticCurve('389a')
}}}

{{{id=36|
L = E.Lseries_dokchitser()
}}}

{{{id=37|
L(1)
///
-1.33174198778018e-19
}}}

{{{id=42|
L(2)
///
0.360092863578881
}}}

{{{id=38|
L(1+I)
///
-0.638409938588039 + 0.715495239204667*I
}}}

{{{id=39|
show(plot(lambda x: abs(L(x)),0, 3), xmin=-0.5, ymin=0, dpi=150)
}}}

{{{id=47|
plot(lambda x: abs(L(x)),0, 3).save('a.eps', xmin=-0.5, ymin=0, dpi=150)
}}}

{{{id=41|
L.taylor_series(1)
///
-2.69129566562797e-23 + (1.52514901968783e-23)*z + 0.759316500288427*z^2 + -0.430302337583362*z^3 + -0.193509313829981*z^4 + 0.459971558373642*z^5 + O(z^6)
}}}

{{{id=44|
L.taylor_series(I)
///
-0.764013101118315 - 9.46601163567108*I + (-19.8890471908356 + 31.2633280771869*I)*z + (67.2680812298021 - 31.3271201136155*I)*z^2 + (-102.731583054904 - 8.43608510668884*I)*z^3 + (93.5537664792735 + 60.7847213025807*I)*z^4 + (-49.4179679165846 - 87.3544330838795*I)*z^5 + O(z^6)
}}}

{{{id=45|
L.derivative(1, 2)
///
1.51863300057685
}}}

{{{id=46|
E = EllipticCurve('389a')
}}}

{{{id=48|
time v = E.Lseries_zeros(20); v
///
[0.000000000, 0.000000000, 2.87609907, 4.41689608, 5.79340263, 6.98596665, 7.47490750, 8.63320525, 9.63307880, 10.3514333, 11.1109355, 11.9335273, 12.6672137, 13.6248537, 15.5056185, 15.9115860, 16.2500699, 17.1798830, 17.8677033, 18.6909039]
CPU time: 0.01 s,  Wall time: 0.76 s
}}}

{{{id=49|
show(list_plot([(1/2, y) for y in v], pointsize=40), xmin=0, figsize=[4,8])
}}}

{{{id=50|
E = EllipticCurve('37a')
}}}

{{{id=51|
L = E.padic_lseries(3)
}}}

{{{id=53|
L.series(4)
///
(O(3^1))*alpha + (O(3^2)) + ((O(3^-1))*alpha + (2*3^-1 + O(3^0)))*T + ((O(3^-1))*alpha + (2*3^-1 + O(3^0)))*T^2 + ((O(3^-2))*alpha + (O(3^-1)))*T^3 + ((O(3^-1))*alpha + (3^-1 + O(3^0)))*T^4 + O(T^5)
}}}

{{{id=58|
latex(L.series(4))
///
\left(2 \cdot 3^{-1} + O(3^{0})\right)T + \left(2 \cdot 3^{-1} + O(3^{0})\right)T^{2} + \left(3^{-1} + O(3^{0})\right)T^{4} + O(\left(1 + O(3^{5})\right)T^{5})
}}}

{{{id=54|
L = E.padic_lseries(5)
}}}

{{{id=55|
L.series(5)
///
O(5^7) + (1 + 4*5 + 2*5^2 + 5^3 + O(5^4))*T + (3 + 3*5^2 + 4*5^3 + O(5^4))*T^2 + (2 + 2*5 + 4*5^2 + 2*5^3 + O(5^4))*T^3 + (4 + 3*5 + 5^2 + 3*5^3 + O(5^4))*T^4 + O(T^5)
}}}

{{{id=56|
latex(L.series(5))
///
O(5^{7}) + \left(1 + 4 \cdot 5 + 2 \cdot 5^{2} + 5^{3} + O(5^{4})\right)T + \left(3 + 3 \cdot 5^{2} + 4 \cdot 5^{3} + O(5^{4})\right)T^{2} + \left(2 + 2 \cdot 5 + 4 \cdot 5^{2} + 2 \cdot 5^{3} + O(5^{4})\right)T^{3} + \left(4 + 3 \cdot 5 + 5^{2} + 3 \cdot 5^{3} + O(5^{4})\right)T^{4} + O(\left(1 + O(5^{9})\right)T^{5})
}}}

{{{id=57|
E = EllipticCurve([1,2,3,4,5])
}}}

{{{id=59|
time E.gens()
///
[(1 : 2 : 1)]
CPU time: 0.01 s,  Wall time: 0.16 s
}}}

{{{id=62|
E = EllipticCurve([12,2007])
}}}

{{{id=63|
time E.gens()
///
[(448569/4096 : -300810003/262144 : 1)]
CPU time: 0.02 s,  Wall time: 0.20 s
}}}

{{{id=64|
time E.simon_two_descent()
///
(1, 1, [(448569/4096 : 300810003/262144 : 1)])
CPU time: 0.05 s,  Wall time: 0.69 s
}}}

{{{id=65|
E.regulator()
///
12.5362247775080
}}}

{{{id=66|
E = EllipticCurve('389a')
}}}

{{{id=67|
E.regulator()
///
0.152460177943144
}}}

{{{id=68|
time E.padic_regulator(5, prec=10)
///
5^2 + 2*5^3 + 2*5^4 + 4*5^5 + 3*5^6 + 4*5^7 + O(5^8)
CPU time: 0.22 s,  Wall time: 0.25 s
}}}

{{{id=69|
time E.padic_regulator(997, prec=10)
///
740*997^2 + 916*997^3 + 472*997^4 + 325*997^5 + 697*997^6 + 642*997^7 + 68*997^8 + 860*997^9 + O(997^10)
CPU time: 0.44 s,  Wall time: 0.45 s
}}}

{{{id=70|
time E.padic_regulator(next_prime(10^5), prec=10)
///
42582*100003^2 + 35250*100003^3 + 12790*100003^4 + 64078*100003^5 + 67283*100003^6 + 48411*100003^7 + 7413*100003^8 + 22370*100003^9 + O(100003^10)
CPU time: 3.95 s,  Wall time: 4.50 s
}}}

{{{id=72|
E = EllipticCurve('37a')
}}}

{{{id=73|
E.Lseries_sympow(2, 16)
///
'2.492262044273650E+00'
}}}

{{{id=74|
sympow('-new_data 2')
///
'Running the new_data script for -sp 2\nMaking the datafiles for -sp 2\n\nRewarping the param_data file\nLeft with 13 entries in param_data\necho \'Removing any old data files\'\ncd datafiles\n/bin/rm -f P02HM.txt P02HS.txt P02HM.bin\n/bin/rm -f P02LM.txt P02LS.txt P02LM.bin\ncd ..\nRemoving any old data files\nRunning the gp script\n\nN=600; dv=0; mx=1;\n\\p 250\nSTR="P02H";\n\\r standard1.gp\nF(k)=if(k%2==0,J(k-2,X)/1!*J(k/2-1,X/2)*sinv(k,X),sqrt(Pi)/2*J(k-1,X)/0!*J(k-2,X)/1!*1/J((k-1)/2,X/2)*two1ms(k,X)*sinv(k,X))\n\\r standard2.gp\n\\l datafiles/P02HM.txt\n\\r standard3.gp\n\\l datafiles/P02HS.txt\ncoeffs(0);\ncoeffE(1);\nSTR="P02L";\n\\r standard1.gp\nF(k)=if(k%2==1,J(k-1,X)/1!*J((k-1)/2,X/2)*sinv(k,X),sqrt(Pi)/2*J(k-1,X)/1!*J(k-1,X)/J(k/2-1,X/2)*two1ms(k,X)*sinv(k,X))\n\\r standard2.gp\n\\l datafiles/P02LM.txt\n\\r standard3.gp\n\\l datafiles/P02LS.txt\ncoeffs(0);\ncoeffO(1);\n\\q\n\necho \'Trimming the data files\'\ncd datafiles\n/bin/grep -v \'^?\' P02HM.txt | /bin/sed \'s/ E/e/\' > .tempfile.123\\\n && echo \'END\' >> .tempfile.123 && mv .tempfile.123 P02HM.txt\n/bin/grep -v \'^?\' P02HS.txt | /bin/sed \'s/ E/e/\' > .tempfile.123\\\n && echo \'END\' >> .tempfile.123 && mv .tempfile.123 P02HS.txt\n/bin/grep -v \'^?\' P02LM.txt | /bin/sed \'s/ E/e/\' > .tempfile.123\\\n && echo \'END\' >> .tempfile.123 && mv .tempfile.123 P02LM.txt\n/bin/grep -v \'^?\' P02LS.txt | /bin/sed \'s/ E/e/\' > .tempfile.123\\\n && echo \'END\' >> .tempfile.123 && mv .tempfile.123 P02LS.txt\necho \'Turning the meshes into binaries\'\nNUM=`/bin/grep -c AT P02HM.txt`\n../sympow -txt2bin $NUM P02HM.bin < P02HM.txt\nNUM=`/bin/grep -c AT P02LM.txt`\n../sympow -txt2bin $NUM P02LM.bin < P02LM.txt\ncd ..\nTrimming the data files\nTurning the meshes into binaries\nRewarping the param_data file\nLeft with 15 entries in param_data\nFinished with -sp 2'
}}}

{{{id=75|
EllipticCurve('37a').modular_degree()
///
2
}}}

{{{id=79|
e = EllipticCurve('37a')
}}}

{{{id=80|
e.simon_two_descent
}}}

{{{id=76|
time EllipticCurve('5077a').modular_degree()
///
1984
CPU time: 0.00 s,  Wall time: 0.01 s
}}}

{{{id=77|
EllipticCurve([1..5]).modular_degree()
///
464
}}}

{{{id=78|
E = EllipticCurve([0, 0, 1, -79, 342])  # a rank 5 curve
time E.modular_degree()
///
33108352
Time: CPU 0.03 s, Wall: 207.52 s
}}}

{{{id=81|
E.conductor()
///
19047851
}}}

{{{id=82|
E = EllipticCurve('37a')
}}}

{{{id=96|
E.sha_an()
///
1
}}}

{{{id=83|
E.analytic_rank()
///
1
}}}

{{{id=84|
E.heegner_discriminants_list(10)
///
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104]
}}}

{{{id=85|
E.heegner_index(-7)       # interval arithmetic
///
[0.99998569 .. 1.0000134]
}}}

{{{id=86|
E.non_surjective()
///
[]
}}}

{{{id=87|
E.shabound_kolyvagin()
///
([2], 1)
}}}

{{{id=88|
E.two_selmer_shabound()   # bound on 2-rank of Sha
///
0
}}}

{{{id=89|
E = EllipticCurve('37b')
}}}

{{{id=90|
E.analytic_rank()
///
0
}}}

{{{id=91|
E.non_surjective()
///
[(3, '3-torsion')]
}}}

{{{id=92|
E.shabound_kato()
///
[2, 3]
}}}

{{{id=94|
E.three_selmer_rank()          # calls magma
///
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/home/was/sd5/bsd/talk/sage_notebook/worksheets/admin/0/code/158.py", line 4, in <module>
    exec compile(ur'E.three_selmer_rank()          # calls magma' + '\n', '', 'single')
  File "/home/was/s/data/extcode/sage/", line 1, in <module>
    
  File "/home/was/s/local/lib/python2.5/site-packages/sage/schemes/elliptic_curves/ell_rational_field.py", line 953, in three_selmer_rank
    raise NotImplementedError, msg[i+1:]
NotImplementedError:  Currently, only the case with irreducible phi3 is implemented.
}}}

{{{id=95|
E.sha_an()
///
1
}}}

{{{id=97|
E.two_selmer_shabound()
///
0
}}}

{{{id=98|
E.padic_lseries(3)
}}}