Properties

Label 248430.ec2
Conductor $248430$
Discriminant $5.635\times 10^{18}$
j-invariant \( \frac{128031684631201}{9922500} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -8695223, 9867524006]) # or
 
sage: E = EllipticCurve("248430ec3")
 
gp: E = ellinit([1, 0, 1, -8695223, 9867524006]) \\ or
 
gp: E = ellinit("248430ec3")
 
magma: E := EllipticCurve([1, 0, 1, -8695223, 9867524006]); // or
 
magma: E := EllipticCurve("248430ec3");
 

\( y^2 + x y + y = x^{3} - 8695223 x + 9867524006 \)

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);
 

\(P\) =  \( \left(\frac{11223}{4}, -\frac{706831}{8}\right) \)\( \left(1740, -3443\right) \)
\(\hat{h}(P)\) ≈  $2.6412329020693663$$1.0970780182329034$

Torsion generators

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

\( \left(\frac{6855}{4}, -\frac{6859}{8}\right) \), \( \left(1691, -846\right) \)

Integral points

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

\( \left(-3405, 1702\right) \), \( \left(-1455, 140152\right) \), \( \left(-1455, -138698\right) \), \( \left(-675, 124552\right) \), \( \left(-675, -123878\right) \), \( \left(795, 58402\right) \), \( \left(795, -59198\right) \), \( \left(1005, 45802\right) \), \( \left(1005, -46808\right) \), \( \left(1587, 7318\right) \), \( \left(1587, -8906\right) \), \( \left(1665, 1702\right) \), \( \left(1665, -3368\right) \), \( \left(1677, 778\right) \), \( \left(1677, -2456\right) \), \( \left(1691, -846\right) \), \( \left(1740, 1702\right) \), \( \left(1740, -3443\right) \), \( \left(1860, 10477\right) \), \( \left(1860, -12338\right) \), \( \left(2220, 37702\right) \), \( \left(2220, -39923\right) \), \( \left(2510, 60852\right) \), \( \left(2510, -63363\right) \), \( \left(6150, 431677\right) \), \( \left(6150, -437828\right) \), \( \left(9972, 951469\right) \), \( \left(9972, -961442\right) \), \( \left(15380, 1866942\right) \), \( \left(15380, -1882323\right) \), \( \left(248700, 123893302\right) \), \( \left(248700, -124142003\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 248430 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(5634682653376822500 \)  =  \(2^{2} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8} \cdot 13^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{128031684631201}{9922500} \)  =  \(2^{-2} \cdot 3^{-4} \cdot 5^{-4} \cdot 7^{-2} \cdot 13^{3} \cdot 3877^{3}\)
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: \(1.80050736928\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.229101442317\)
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: \( 512 \)  = \( 2\cdot2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{2} \)
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 248430.2.a.ec

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);
 

\( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} + q^{15} + q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 14155776
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 13.1999627266 \)

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\((j)_{-}\)
\(2\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(3\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(5\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(7\) \(4\) \( I_2^{*} \) Additive -1 2 8 2
\(13\) \(4\) \( I_0^{*} \) Additive 1 2 6 0

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 X98.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 6 \\ 4 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 7 \end{array}\right)$ and has index 24.

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.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 248430.ec 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
$2$ \(\Q(\sqrt{91}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{14}, \sqrt{-26})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-14}, \sqrt{26})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{15}, \sqrt{91})\) \(\Z/2\Z \times \Z/8\Z\) 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.