Properties

Label 489762.dg2
Conductor $489762$
Discriminant $6.718\times 10^{19}$
j-invariant \( \frac{614716917569296417}{19093020912} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -26943026, -53820986943]) # or
 
sage: E = EllipticCurve("489762dg4")
 
gp: E = ellinit([1, -1, 1, -26943026, -53820986943]) \\ or
 
gp: E = ellinit("489762dg4")
 
magma: E := EllipticCurve([1, -1, 1, -26943026, -53820986943]); // or
 
magma: E := EllipticCurve("489762dg4");
 

\( y^2 + x y + y = x^{3} - x^{2} - 26943026 x - 53820986943 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(98433, -30887661\right) \)\( \left(6497, -216189\right) \)
\(\hat{h}(P)\) ≈  $4.377203101841202$$4.8016607007523895$

Torsion generators

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

\( \left(-\frac{12037}{4}, \frac{12033}{8}\right) \)

Integral points

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

\( \left(-3007, 2187\right) \), \( \left(-3007, 819\right) \), \( \left(6497, 209691\right) \), \( \left(6497, -216189\right) \), \( \left(98433, 30789227\right) \), \( \left(98433, -30887661\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 489762 \)  =  \(2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \cdot 23\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(67183448212742530032 \)  =  \(2^{4} \cdot 3^{8} \cdot 7^{8} \cdot 13^{6} \cdot 23 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{614716917569296417}{19093020912} \)  =  \(2^{-4} \cdot 3^{-2} \cdot 7^{-8} \cdot 23^{-1} \cdot 850273^{3}\)
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: \(20.0255784321736\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.0662656973658883\)
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: \( 64 \)  = \( 2^{2}\cdot2\cdot2\cdot2^{2}\cdot1 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(2\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 489762.2.a.dg

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);
 

\( q + q^{2} + q^{4} - 2q^{5} - q^{7} + q^{8} - 2q^{10} - 4q^{11} - q^{14} + q^{16} + 6q^{17} - 4q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 37748736
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1 (conditional*)
* The Manin constant is correct provided that curve 489762.dg5 is optimal.

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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 21.232142719412362 \)

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\((j)_{-}\)
\(2\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(3\) \(2\) \( I_2^{*} \) Additive -1 2 8 2
\(7\) \(2\) \( I_{8} \) Non-split multiplicative 1 1 8 8
\(13\) \(4\) \( I_0^{*} \) Additive 1 2 6 0
\(23\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1

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 X102.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right)$ 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
\(2\) 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, 4 and 8.
Its isogeny class 489762.dg consists of 6 curves linked by isogenies of degrees dividing 8.