# Properties

 Label 485100.he4 Conductor $485100$ Discriminant $-1.319\times 10^{26}$ j-invariant $$\frac{7549996227362816}{6152409907875}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 113542800, 297499292125])

gp: E = ellinit([0, 0, 0, 113542800, 297499292125])

magma: E := EllipticCurve([0, 0, 0, 113542800, 297499292125]);

$$y^2=x^3+113542800x+297499292125$$

## Mordell-Weil group structure

$$\Z\times \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-1945, 263250\right)$$ $$\hat{h}(P)$$ ≈ $4.3478802688982825320695631187$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-2485, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-2485, 0\right)$$, $$(-1945,\pm 263250)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$485100$$ = $$2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-131917083150101525718750000$$ = $$-1 \cdot 2^{4} \cdot 3^{10} \cdot 5^{9} \cdot 7^{9} \cdot 11^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{7549996227362816}{6152409907875}$$ = $$2^{20} \cdot 3^{-4} \cdot 5^{-3} \cdot 7^{-3} \cdot 11^{-6} \cdot 1931^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$3.6998273143448474928122000689\dots$$ Stable Faltings height: $$1.1417980790794373707891107050\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$4.3478802688982825320695631187\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.037743834849732509186483968602\dots$$ 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: $$192$$  = $$1\cdot2^{2}\cdot2^{2}\cdot2\cdot( 2 \cdot 3 )$$ 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: $$1$$ (exact)

## Modular invariants

Modular form 485100.2.a.he

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);

$$q + q^{11} + 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})$$

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 143327232 $$\Gamma_0(N)$$-optimal: no Manin constant: 1 (conditional*)
* The Manin constant is correct provided that curve 485100.he3 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);

$$L'(E,1)$$ ≈ $$7.8770723911539527668260247738014698175$$

## Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

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$$(j)_{-}$$
$$2$$ $$1$$ $$IV$$ Additive -1 2 4 0
$$3$$ $$4$$ $$I_4^{*}$$ Additive -1 2 10 4
$$5$$ $$4$$ $$I_3^{*}$$ Additive 1 2 9 3
$$7$$ $$2$$ $$I_3^{*}$$ Additive -1 2 9 3
$$11$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6

## 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 X6.

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

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
$$3$$ B

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 485100.he consists of 4 curves linked by isogenies of degrees dividing 6.