Properties

Label 57330.bu5
Conductor $57330$
Discriminant $1.305\times 10^{15}$
j-invariant \( \frac{30400540561}{15210000} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -28674, 694980]) # or
 
sage: E = EllipticCurve("57330cm2")
 
gp: E = ellinit([1, -1, 0, -28674, 694980]) \\ or
 
gp: E = ellinit("57330cm2")
 
magma: E := EllipticCurve([1, -1, 0, -28674, 694980]); // or
 
magma: E := EllipticCurve("57330cm2");
 

\( y^2 + x y = x^{3} - x^{2} - 28674 x + 694980 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-159, 1182\right) \)\( \left(-144, 1422\right) \)
\(\hat{h}(P)\) ≈  $1.2820868472649247$$1.5564877937547175$

Torsion generators

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

\( \left(156, -78\right) \), \( \left(\frac{99}{4}, -\frac{99}{8}\right) \)

Integral points

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

\( \left(-180, 90\right) \), \( \left(-159, 1182\right) \), \( \left(-159, -1023\right) \), \( \left(-144, 1422\right) \), \( \left(-144, -1278\right) \), \( \left(-89, 1637\right) \), \( \left(-89, -1548\right) \), \( \left(-36, 1314\right) \), \( \left(-36, -1278\right) \), \( \left(-24, 1182\right) \), \( \left(-24, -1158\right) \), \( \left(9, 657\right) \), \( \left(9, -666\right) \), \( \left(16, 482\right) \), \( \left(16, -498\right) \), \( \left(156, -78\right) \), \( \left(171, 792\right) \), \( \left(171, -963\right) \), \( \left(181, 1097\right) \), \( \left(181, -1278\right) \), \( \left(184, 1182\right) \), \( \left(184, -1366\right) \), \( \left(261, 3177\right) \), \( \left(261, -3438\right) \), \( \left(445, 8490\right) \), \( \left(445, -8935\right) \), \( \left(576, 12942\right) \), \( \left(576, -13518\right) \), \( \left(781, 20922\right) \), \( \left(781, -21703\right) \), \( \left(1731, 70797\right) \), \( \left(1731, -72528\right) \), \( \left(1920, 82830\right) \), \( \left(1920, -84750\right) \), \( \left(3096, 170442\right) \), \( \left(3096, -173538\right) \), \( \left(27036, 4431762\right) \), \( \left(27036, -4458798\right) \), \( \left(42291, 8675757\right) \), \( \left(42291, -8718048\right) \), \( \left(672856, 551592222\right) \), \( \left(672856, -552265078\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 57330 \)  =  \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(1304502700410000 \)  =  \(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{6} \cdot 13^{2} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{30400540561}{15210000} \)  =  \(2^{-4} \cdot 3^{-2} \cdot 5^{-4} \cdot 13^{-2} \cdot 3121^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1.11690826578\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.427669440563\)
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: \( 256 \)  = \( 2\cdot2^{2}\cdot2^{2}\cdot2^{2}\cdot2 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(4\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 57330.2.a.bu

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

\( q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - 4q^{11} - q^{13} + q^{16} - 6q^{17} - 4q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 393216
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

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

\( L^{(2)}(E,1)/2! \) ≈ \( 7.64268053101 \)

Local data

This elliptic curve is not semistable.

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(2\) \( I_{4} \) Non-split multiplicative 1 1 4 4
\(3\) \(4\) \( I_2^{*} \) Additive -1 2 8 2
\(5\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(7\) \(4\) \( I_0^{*} \) 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 X25.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right)$ and has index 12.

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

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

$p$-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.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit add split add ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 6 - 3 - 2 2 2 2 2 2 2 2 2 2 2,2
$\mu$-invariant(s) 0 - 0 - 0 0 0 0 0 0 0 0 0 0 0,0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 57330.bu consists of 6 curves linked by isogenies of degrees dividing 8.

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
$2$ \(\Q(\sqrt{21}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-21}, \sqrt{39})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-21}, \sqrt{-39})\) \(\Z/2\Z \times \Z/4\Z\) 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.