# Properties

 Label 46410.be4 Conductor 46410 Discriminant 580277967360000 j-invariant $$\frac{11650256451486052494789241}{580277967360000}$$ 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, -4722848, -3950910994]) # or

sage: E = EllipticCurve("46410be3")

gp: E = ellinit([1, 0, 1, -4722848, -3950910994]) \\ or

gp: E = ellinit("46410be3")

magma: E := EllipticCurve([1, 0, 1, -4722848, -3950910994]); // or

magma: E := EllipticCurve("46410be3");

$$y^2 + x y + y = x^{3} - 4722848 x - 3950910994$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(\frac{769745}{121}, -\frac{633418263}{1331}\right)$$ $$\hat{h}(P)$$ ≈ 10.04495921116066

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-1255, 627\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1255, 627\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$46410$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$580277967360000$$ = $$2^{12} \cdot 3 \cdot 5^{4} \cdot 7 \cdot 13^{3} \cdot 17^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{11650256451486052494789241}{580277967360000}$$ = $$2^{-12} \cdot 3^{-1} \cdot 5^{-4} \cdot 7^{-1} \cdot 13^{-3} \cdot 17^{-3} \cdot 71^{3} \cdot 1583^{3} \cdot 2017^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$10.0449592112$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.102411464077$$ 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: $$24$$  = $$2\cdot1\cdot2^{2}\cdot1\cdot3\cdot1$$ 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 46410.2.a.be

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);

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + q^{7} - q^{8} + q^{9} - q^{10} + q^{12} + q^{13} - q^{14} + q^{15} + q^{16} - q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 912384 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### 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);

$$L'(E,1)$$ ≈ $$6.17231387648$$

## Local data

This elliptic curve is 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{12}$$ Non-split multiplicative 1 1 12 12
$$3$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$5$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$7$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$13$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$17$$ $$1$$ $$I_{3}$$ Non-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.

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.1.2

## $p$-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 split split split ss split nonsplit ordinary ss ordinary ordinary ordinary ordinary ordinary ss 7 2 2 2 1,1 2 1 1 1,1 3 1 1 1 1 1,1 0 1 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, 3, 4, 6 and 12.
Its isogeny class 46410.be consists of 8 curves linked by isogenies of degrees dividing 12.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-3})$$ $$\Z/6\Z$$ Not in database
$$\Q(\sqrt{-51})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-91})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{4641})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
3 3.1.297675.3 $$\Z/6\Z$$ Not in database
4 $$\Q(\sqrt{-3}, \sqrt{17})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{-91})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-51}, \sqrt{-91})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{-1547})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
6 6.0.265831216875.4 $$\Z/3\Z \times \Z/6\Z$$ Not in database
$$x^{6} - 3 x^{5} + 42 x^{4} + 131 x^{3} + 231 x^{2} - 8382 x + 17212$$ $$\Z/12\Z$$ Not in database
$$x^{6} - 3 x^{5} + 72 x^{4} + 71 x^{3} + 1341 x^{2} - 15762 x + 30332$$ $$\Z/12\Z$$ Not in database
$$x^{6} - 3 x^{5} - 3477 x^{4} + 6749 x^{3} + 4033635 x^{2} - 4767915 x - 1560519470$$ $$\Z/2\Z \times \Z/6\Z$$ 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.