Properties

 Label 51870.d4 Conductor 51870 Discriminant -172623360000 j-invariant $$\frac{73197245859191}{172623360000}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, 872, 17728]) # or

sage: E = EllipticCurve("51870a1")

gp: E = ellinit([1, 1, 0, 872, 17728]) \\ or

gp: E = ellinit("51870a1")

magma: E := EllipticCurve([1, 1, 0, 872, 17728]); // or

magma: E := EllipticCurve("51870a1");

$$y^2 + x y = x^{3} + x^{2} + 872 x + 17728$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(9, -167\right)$$ $$\left(609, 14758\right)$$ $$\hat{h}(P)$$ ≈ 0.9281194288485692 4.832762561612888

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-16, 8\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-16, 8\right)$$, $$\left(-7, 110\right)$$, $$\left(-7, -103\right)$$, $$\left(9, 158\right)$$, $$\left(9, -167\right)$$, $$\left(48, 392\right)$$, $$\left(48, -440\right)$$, $$\left(609, 14758\right)$$, $$\left(609, -15367\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$51870$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-172623360000$$ = $$-1 \cdot 2^{12} \cdot 3 \cdot 5^{4} \cdot 7 \cdot 13^{2} \cdot 19$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{73197245859191}{172623360000}$$ = $$2^{-12} \cdot 3^{-1} \cdot 5^{-4} \cdot 7^{-1} \cdot 13^{-2} \cdot 19^{-1} \cdot 59^{3} \cdot 709^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$4.17580103891$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.708291160443$$ 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: $$8$$  = $$2\cdot1\cdot2\cdot1\cdot2\cdot1$$ 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: $$1$$ (rounded)

Modular invariants

Modular form 51870.2.a.d

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 61440 $$\Gamma_0(N)$$-optimal: yes 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);

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

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
$$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$5$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$7$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$13$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$19$$ $$1$$ $$I_{1}$$ 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 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

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ of good ordinary reduction.

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 $$\Q(\sqrt{-3})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-399})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{133})$$ $$\Z/4\Z$$ Not in database
4 $$\Q(\sqrt{-3}, \sqrt{133})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.