# Properties

 Label 46410b2 Conductor 46410 Discriminant 15197834433600 j-invariant $$\frac{127787213284071769}{15197834433600}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, -10493, 364413]); // or

magma: E := EllipticCurve("46410b2");

sage: E = EllipticCurve([1, 1, 0, -10493, 364413]) # or

sage: E = EllipticCurve("46410b2")

gp: E = ellinit([1, 1, 0, -10493, 364413]) \\ or

gp: E = ellinit("46410b2")

$$y^2 + x y = x^{3} + x^{2} - 10493 x + 364413$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(91, -462\right)$$ $$\left(-79, 881\right)$$ $$\hat{h}(P)$$ ≈ 1.2229946441 2.51419662016

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(42, -21\right)$$, $$\left(74, -37\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-86, 843\right)$$, $$\left(-79, 881\right)$$, $$\left(-62, 915\right)$$, $$\left(-7, 665\right)$$, $$\left(29, 278\right)$$, $$\left(34, 203\right)$$, $$\left(42, -21\right)$$, $$\left(74, -37\right)$$, $$\left(91, 371\right)$$, $$\left(123, 915\right)$$, $$\left(146, 1331\right)$$, $$\left(159, 1578\right)$$, $$\left(434, 8603\right)$$, $$\left(679, 17178\right)$$, $$\left(10442, 1061819\right)$$, $$\left(55154, 12925403\right)$$

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

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$46410$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$15197834433600$$ = $$2^{6} \cdot 3^{4} \cdot 5^{2} \cdot 7^{4} \cdot 13^{2} \cdot 17^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{127787213284071769}{15197834433600}$$ = $$2^{-6} \cdot 3^{-4} \cdot 5^{-2} \cdot 7^{-4} \cdot 13^{-2} \cdot 17^{-2} \cdot 109^{3} \cdot 4621^{3}$$ 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: $$1.97492848297$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.676447010267$$ 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: $$64$$  = $$2\cdot2\cdot2\cdot2\cdot2\cdot2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$4$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 46410.2.a.a

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 147456 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### 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])

$$L^{(2)}(E,1)/2!$$ ≈ $$5.34373787119$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$3$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$7$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$13$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$17$$ $$2$$ $$I_{2}$$ 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 X8c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 6 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 5 \end{array}\right)$ and has index 12.

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
$$2$$ 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.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 nonsplit nonsplit nonsplit nonsplit ordinary nonsplit split ss ordinary ordinary ss ordinary ordinary ordinary ss 5 2 4 2 2 2 3 2,2 2 2 2,2 2 2 2 2,2 1 0 0 0 0 0 0 0,0 0 0 0,0 0 0 0 0,0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 46410b consists of 4 curves linked by isogenies of degrees dividing 4.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 $$\Q(\sqrt{-13}, \sqrt{-85})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{2}, \sqrt{85})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-2}, \sqrt{13})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.