# Properties

 Label 436800hy3 Conductor 436800 Discriminant 89680698351738224640000000 j-invariant $$\frac{443915739051786565201}{21894701746029840}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 0, -254280033, 1492786151937]); // or

magma: E := EllipticCurve("436800hy3");

sage: E = EllipticCurve([0, -1, 0, -254280033, 1492786151937]) # or

sage: E = EllipticCurve("436800hy3")

gp: E = ellinit([0, -1, 0, -254280033, 1492786151937]) \\ or

gp: E = ellinit("436800hy3")

$$y^2 = x^{3} - x^{2} - 254280033 x + 1492786151937$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-1173, -1337700\right)$$ $$\left(65712, -16386825\right)$$ $$\hat{h}(P)$$ ≈ 0.762861604658 1.16750162096

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(-18323, 0\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-18323, 0\right)$$, $$\left(-13913, 1528800\right)$$, $$\left(-9363, 1747200\right)$$, $$\left(-1173, 1337700\right)$$, $$\left(5687, 480200\right)$$, $$\left(7277, 166400\right)$$, $$\left(7563, 46956\right)$$, $$\left(11112, 197925\right)$$, $$\left(13037, 627200\right)$$, $$\left(15312, 1090425\right)$$, $$\left(25581, 3424512\right)$$, $$\left(65712, 16386825\right)$$, $$\left(541677, 398496000\right)$$, $$\left(292507512, 5002712375325\right)$$

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$436800$$ = $$2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$89680698351738224640000000$$ = $$2^{22} \cdot 3^{2} \cdot 5^{7} \cdot 7^{12} \cdot 13^{3}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{443915739051786565201}{21894701746029840}$$ = $$2^{-4} \cdot 3^{-2} \cdot 5^{-1} \cdot 7^{-12} \cdot 11^{3} \cdot 13^{-3} \cdot 37^{3} \cdot 18743^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$0.820684922639146$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0596269372576519$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$1152$$  = $$2^{2}\cdot2\cdot2^{2}\cdot( 2^{2} \cdot 3 )\cdot3$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 436800.2.a.hy

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q - q^{3} + q^{7} + q^{9} + q^{13} - 6q^{17} - 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 127401984 $$\Gamma_0(N)$$-optimal: unknown* (one of 6 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 436800hy1 is optimal.

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

$$L^{(2)}(E,1)/2!$$ ≈ $$14.093259376465523$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(E)[5]

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$4$$ $$I_12^{*}$$ Additive -1 6 22 4
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$5$$ $$4$$ $$I_1^{*}$$ Additive 1 2 7 1
$$7$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$13$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3

## 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 X13.

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

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

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(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, 3, 4, 6 and 12.
Its isogeny class 436800hy consists of 8 curves linked by isogenies of degrees dividing 12.