# Properties

 Label 489762.dg4 Conductor $489762$ Discriminant $9.267\times 10^{19}$ j-invariant $$\frac{169967019783457}{26337394944}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -1755266, -765489279]) # or

sage: E = EllipticCurve("489762dg2")

gp: E = ellinit([1, -1, 1, -1755266, -765489279]) \\ or

gp: E = ellinit("489762dg2")

magma: E := EllipticCurve([1, -1, 1, -1755266, -765489279]); // or

magma: E := EllipticCurve("489762dg2");

$$y^2 + x y + y = x^{3} - x^{2} - 1755266 x - 765489279$$

## 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);

 $$P$$ = $$\left(1557, 15783\right)$$ $$\left(2909, 135435\right)$$ $$\hat{h}(P)$$ ≈ $2.188601550920601$ $2.4008303503761947$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(1505, -753\right)$$, $$\left(-991, 495\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-991, 495\right)$$, $$\left(-979, 4215\right)$$, $$\left(-979, -3237\right)$$, $$\left(-795, 11667\right)$$, $$\left(-795, -10873\right)$$, $$\left(-523, 3303\right)$$, $$\left(-523, -2781\right)$$, $$\left(1505, -753\right)$$, $$\left(1557, 15783\right)$$, $$\left(1557, -17341\right)$$, $$\left(2909, 135435\right)$$, $$\left(2909, -138345\right)$$, $$\left(5093, 347283\right)$$, $$\left(5093, -352377\right)$$, $$\left(8417, 757839\right)$$, $$\left(8417, -766257\right)$$, $$\left(24609, 3842415\right)$$, $$\left(24609, -3867025\right)$$, $$\left(98381, 30805815\right)$$, $$\left(98381, -30904197\right)$$

## 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: $$92674544140192944384$$ = $$2^{8} \cdot 3^{10} \cdot 7^{4} \cdot 13^{6} \cdot 23^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{169967019783457}{26337394944}$$ = $$2^{-8} \cdot 3^{-4} \cdot 7^{-4} \cdot 13^{3} \cdot 23^{-2} \cdot 4261^{3}$$ 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: $$5.00639460804339$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.132531394731777$$ 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: $$512$$  = $$2^{3}\cdot2^{2}\cdot2\cdot2^{2}\cdot2$$ 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 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 18874368 $$\Gamma_0(N)$$-optimal: unknown* (one of 4 curves in this isogeny class which might be optimal) Manin constant: 1 (conditional*)
* The optimal curve in each isogeny class has not been determined in all cases for conductors over 400000. 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$$(j)_{-}$$
$$2$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8
$$3$$ $$4$$ $$I_4^{*}$$ Additive -1 2 10 4
$$7$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$13$$ $$4$$ $$I_0^{*}$$ Additive 1 2 6 0
$$23$$ $$2$$ $$I_{2}$$ 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 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 489762.dg consists of 6 curves linked by isogenies of degrees dividing 8.