# Properties

 Label 25872d2 Conductor 25872 Discriminant 131194635264 j-invariant $$\frac{3650692}{1089}$$ 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([0, -1, 0, -1584, 17424]); // or

magma: E := EllipticCurve("25872d2");

sage: E = EllipticCurve([0, -1, 0, -1584, 17424]) # or

sage: E = EllipticCurve("25872d2")

gp: E = ellinit([0, -1, 0, -1584, 17424]) \\ or

gp: E = ellinit("25872d2")

$$y^2 = x^{3} - x^{2} - 1584 x + 17424$$

## 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()

 $$P$$ = $$\left(-42, 90\right)$$ $$\left(-16, 196\right)$$ $$\hat{h}(P)$$ ≈ 2.92691651819 0.949438296678

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(12, 0\right)$$, $$\left(33, 0\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-44, 0\right)$$, $$(-42,\pm 90)$$, $$(-16,\pm 196)$$, $$(0,\pm 132)$$, $$(5,\pm 98)$$, $$(6,\pm 90)$$, $$\left(12, 0\right)$$, $$\left(33, 0\right)$$, $$(37,\pm 90)$$, $$(44,\pm 176)$$, $$(54,\pm 294)$$, $$(110,\pm 1078)$$, $$(180,\pm 2352)$$, $$(208,\pm 2940)$$, $$(1364,\pm 50336)$$, $$(2112,\pm 97020)$$, $$(35334,\pm 6641754)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$25872$$ = $$2^{4} \cdot 3 \cdot 7^{2} \cdot 11$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$131194635264$$ = $$2^{10} \cdot 3^{2} \cdot 7^{6} \cdot 11^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{3650692}{1089}$$ = $$2^{2} \cdot 3^{-2} \cdot 11^{-2} \cdot 97^{3}$$ 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: $$1.6110777923$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.96531687335$$ 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: $$64$$  = $$2^{2}\cdot2\cdot2^{2}\cdot2$$ 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 25872.2.a.h

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)

$$q - q^{3} - 2q^{5} + q^{9} - q^{11} - 2q^{13} + 2q^{15} - 6q^{17} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 24576 $$\Gamma_0(N)$$-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])

$$L^{(2)}(E,1)/2!$$ ≈ $$6.22080230875$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$4$$ $$I_2^{*}$$ Additive 1 4 10 0
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$7$$ $$4$$ $$I_0^{*}$$ Additive -1 2 6 0
$$11$$ $$2$$ $$I_{2}$$ Non-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 X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by  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
$$2$$ 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 add nonsplit ordinary add nonsplit ordinary ordinary ss ordinary ordinary ss ordinary ordinary ordinary ordinary - 2 2 - 2 4 2 2,2 2 2 2,2 2 2 2 2 - 0 0 - 0 0 0 0,0 0 0 0,0 0 0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 25872d 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 $$\Q(\sqrt{-6}, \sqrt{14})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{21}, \sqrt{33})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-14}, \sqrt{22})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.