# Properties

 Label 465690.o2 Conductor $465690$ Discriminant $-2.446\times 10^{20}$ j-invariant $$-\frac{119305480789133569}{5200091136000}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -3702424, 2843141222])

gp: E = ellinit([1, 0, 1, -3702424, 2843141222])

magma: E := EllipticCurve([1, 0, 1, -3702424, 2843141222]);

$$y^2+xy+y=x^3-3702424x+2843141222$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(1018, 10862\right)$$ $$\hat{h}(P)$$ ≈ $2.2990569512289826942096525088$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-2231, 1115\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-2231, 1115\right)$$, $$\left(1018, 10862\right)$$, $$\left(1018, -11881\right)$$, $$\left(1225, 11483\right)$$, $$\left(1225, -12709\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$465690$$ = $$2 \cdot 3 \cdot 5 \cdot 19^{2} \cdot 43$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-244642868773410816000$$ = $$-1 \cdot 2^{14} \cdot 3^{10} \cdot 5^{3} \cdot 19^{6} \cdot 43$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{119305480789133569}{5200091136000}$$ = $$-1 \cdot 2^{-14} \cdot 3^{-10} \cdot 5^{-3} \cdot 7^{3} \cdot 43^{-1} \cdot 70327^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$2.6776360889661848560180141437\dots$$ Stable Faltings height: $$1.2054165993829646260135004278\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$2.2990569512289826942096525088\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.17402933743988812566978518422\dots$$ 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: $$40$$  = $$2\cdot( 2 \cdot 5 )\cdot1\cdot2\cdot1$$ 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: $$1$$ (exact)

## Modular invariants

Modular form 465690.2.a.o

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);

$$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} - q^{8} + q^{9} + q^{10} - 4q^{11} + q^{12} - 4q^{13} + 4q^{14} - q^{15} + q^{16} + 4q^{17} - q^{18} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 34836480 $$\Gamma_0(N)$$-optimal: not computed* (one of 2 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 this curve 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);

$$L'(E,1)$$ ≈ $$4.0010335795894904654198749173357357259$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{14}$$ Non-split multiplicative 1 1 14 14
$$3$$ $$10$$ $$I_{10}$$ Split multiplicative -1 1 10 10
$$5$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3
$$19$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0
$$43$$ $$1$$ $$I_{1}$$ Non-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 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

## $p$-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.
Its isogeny class 465690.o consists of 2 curves linked by isogenies of degree 2.