# Properties

 Label 51870.d1 Conductor 51870 Discriminant 6345626621880 j-invariant $$\frac{141369383441705190409}{6345626621880}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, -108528, 13715688]); // or
magma: E := EllipticCurve("51870a4");
sage: E = EllipticCurve([1, 1, 0, -108528, 13715688]) # or
sage: E = EllipticCurve("51870a4")
gp: E = ellinit([1, 1, 0, -108528, 13715688]) \\ or
gp: E = ellinit("51870a4")

$$y^2 + x y = x^{3} + x^{2} - 108528 x + 13715688$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(203, -420\right)$$ $$\left(463, 7705\right)$$ $$\hat{h}(P)$$ ≈ 0.928119428849 4.6480840941

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$$\left(\frac{763}{4}, -\frac{763}{8}\right)$$

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$$\left(193, -23\right)$$, $$\left(203, 217\right)$$, $$\left(281, 2206\right)$$, $$\left(463, 7705\right)$$, $$\left(2447, 118809\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$51870$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$6345626621880$$ = $$2^{3} \cdot 3 \cdot 5 \cdot 7^{4} \cdot 13^{2} \cdot 19^{4}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{141369383441705190409}{6345626621880}$$ = $$2^{-3} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-4} \cdot 11^{3} \cdot 13^{-2} \cdot 19^{-4} \cdot 473579^{3}$$ 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: $$4.17580103891$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$0.708291160443$$ 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: $$8$$  = $$1\cdot1\cdot1\cdot2\cdot2\cdot2$$ 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 51870.2.a.d

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)

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 245760 $$\Gamma_0(N)$$-optimal: no Manin constant: not computed

#### 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])

$$L^{(2)}(E,1)/2!$$ ≈ $$5.91536592726$$

## 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3
$$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$7$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$13$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$19$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4

## 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

## $p$-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 ss split ordinary nonsplit ordinary ordinary ss ordinary ordinary ordinary ordinary 4 2 2 4 2,2 5 2 4 2 2 2,2 2 2 2 2 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 and 4.
Its isogeny class 51870.d 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{30})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{3})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{10})$$ $$\Z/4\Z$$ Not in database
4 $$\Q(\sqrt{3}, \sqrt{10})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.