Properties

Label 330330cr2
Conductor 330330
Discriminant 7796397117189681000000
j-invariant \( \frac{7850236389974007121}{4400862921000000} \)
CM no
Rank 2
Torsion Structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -5010008, 762943718]); // or
 
magma: E := EllipticCurve("330330cr2");
 
sage: E = EllipticCurve([1, 0, 1, -5010008, 762943718]) # or
 
sage: E = EllipticCurve("330330cr2")
 
gp: E = ellinit([1, 0, 1, -5010008, 762943718]) \\ or
 
gp: E = ellinit("330330cr2")
 

\( y^2 + x y + y = x^{3} - 5010008 x + 762943718 \)

Mordell-Weil group structure

\(\Z^2 \times \Z/{2}\Z \times \Z/{2}\Z\)

Infinite order Mordell-Weil generators and heights

magma: Generators(E);
 
sage: E.gens()
 

\(P\) =  \( \left(26289, -4260245\right) \)\( \left(18864, -2582195\right) \)
\(\hat{h}(P)\) ≈  0.8742540457691.1724658223

Torsion generators

magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 

\( \left(\frac{8631}{4}, -\frac{8635}{8}\right) \), \( \left(153, -77\right) \)

Integral points

magma: IntegralPoints(E);
 
sage: E.integral_points()
 

\( \left(-2311, 1155\right) \), \( \left(-1926, 58135\right) \), \( \left(-1926, -56210\right) \), \( \left(-1791, 64075\right) \), \( \left(-1791, -62285\right) \), \( \left(-936, 68530\right) \), \( \left(-936, -67595\right) \), \( \left(-738, 64075\right) \), \( \left(-738, -63338\right) \), \( \left(-231, 43795\right) \), \( \left(-231, -43565\right) \), \( \left(-111, 36355\right) \), \( \left(-111, -36245\right) \), \( \left(-36, 30730\right) \), \( \left(-36, -30695\right) \), \( \left(153, -77\right) \), \( \left(2169, 8995\right) \), \( \left(2169, -11165\right) \), \( \left(2178, 12478\right) \), \( \left(2178, -14657\right) \), \( \left(2529, 64075\right) \), \( \left(2529, -66605\right) \), \( \left(2694, 81235\right) \), \( \left(2694, -83930\right) \), \( \left(4124, 222090\right) \), \( \left(4124, -226215\right) \), \( \left(4689, 281155\right) \), \( \left(4689, -285845\right) \), \( \left(4914, 305455\right) \), \( \left(4914, -310370\right) \), \( \left(5697, 393547\right) \), \( \left(5697, -399245\right) \), \( \left(6714, 516055\right) \), \( \left(6714, -522770\right) \), \( \left(17049, 2198515\right) \), \( \left(17049, -2215565\right) \), \( \left(18864, 2563330\right) \), \( \left(18864, -2582195\right) \), \( \left(26289, 4233955\right) \), \( \left(26289, -4260245\right) \), \( \left(39192, 7726642\right) \), \( \left(39192, -7765835\right) \), \( \left(145809, 55597675\right) \), \( \left(145809, -55743485\right) \), \( \left(444564, 296189905\right) \), \( \left(444564, -296634470\right) \), \( \left(798489, 713113555\right) \), \( \left(798489, -713912045\right) \), \( \left(6899409, 18119038315\right) \), \( \left(6899409, -18125937725\right) \)

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 330330 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \cdot 13\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(7796397117189681000000 \)  =  \(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 7^{2} \cdot 11^{6} \cdot 13^{2} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{7850236389974007121}{4400862921000000} \)  =  \(2^{-6} \cdot 3^{-12} \cdot 5^{-6} \cdot 7^{-2} \cdot 13^{-2} \cdot 31^{3} \cdot 61^{3} \cdot 1051^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(2\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(0.874350249798\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.113704107973\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 2304 \)  = \( 2\cdot( 2^{2} \cdot 3 )\cdot( 2 \cdot 3 )\cdot2\cdot2^{2}\cdot2 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(4\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 330330.2.a.cr

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

\( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + q^{12} - q^{13} + q^{14} + q^{15} + q^{16} - 6q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 26542080
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L^{(2)}(E,1)/2! \) ≈ \( 14.3160789901 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(2\) \( I_{6} \) Non-split multiplicative 1 1 6 6
\(3\) \(12\) \( I_{12} \) Split multiplicative -1 1 12 12
\(5\) \(6\) \( I_{6} \) Split multiplicative -1 1 6 6
\(7\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(11\) \(4\) \( I_0^{*} \) Additive -1 2 6 0
\(13\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $$ and has index 6.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(2\) Cs
\(3\) B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 330330cr consists of 8 curves linked by isogenies of degrees dividing 12.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{-11}) \) \(\Z/2\Z \times \Z/6\Z\) Not in database
4 \(\Q(\sqrt{11}, \sqrt{65})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{-11}, \sqrt{-14})\) \(\Z/2\Z \times \Z/12\Z\) Not in database
\(\Q(\sqrt{-154}, \sqrt{-715})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
6 6.2.2464378373457.3 \(\Z/2\Z \times \Z/6\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.