# Properties

 Label 53312ca1 Conductor $53312$ Discriminant $-3.294\times 10^{14}$ j-invariant $$-\frac{728871512656}{410338673}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -17425, 1249361]) # or

sage: E = EllipticCurve("53312ca1")

gp: E = ellinit([0, -1, 0, -17425, 1249361]) \\ or

gp: E = ellinit("53312ca1")

magma: E := EllipticCurve([0, -1, 0, -17425, 1249361]); // or

magma: E := EllipticCurve("53312ca1");

$$y^2 = x^{3} - x^{2} - 17425 x + 1249361$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-5, 1156\right)$$ $$\left(41, 776\right)$$ $$\hat{h}(P)$$ ≈ $0.35977293019082124$ $3.5713242610146727$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-5,\pm 1156)$$, $$(41,\pm 776)$$, $$(97,\pm 680)$$, $$(233,\pm 3128)$$, $$(455,\pm 9344)$$, $$(539,\pm 12172)$$, $$(1729,\pm 71672)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$53312$$ = $$2^{6} \cdot 7^{2} \cdot 17$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-329426452103168$$ = $$-1 \cdot 2^{14} \cdot 7^{2} \cdot 17^{7}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{728871512656}{410338673}$$ = $$-1 \cdot 2^{4} \cdot 7 \cdot 17^{-7} \cdot 1867^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential 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: $$0.395505134016$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.5028010907$$ 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: $$28$$  = $$2^{2}\cdot1\cdot7$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

Modular form 53312.2.a.t

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);

$$q - q^{3} - 4q^{5} - 2q^{9} - 3q^{11} - 5q^{13} + 4q^{15} + q^{17} - 6q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 258048 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$5.5680915573$$

## Local data

This elliptic curve is not 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_4^{*}$$ Additive -1 6 14 0
$$7$$ $$1$$ $$II$$ Additive -1 2 2 0
$$17$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 $$p$$ .

## $p$-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 $$p\ge 5$$ 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 add ordinary ordinary add ordinary ordinary split ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary - 2 4 - 2 4 3 2 4 2 2,2 2 2 2 2 - 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 no rational isogenies. Its isogeny class 53312ca consists of this curve only.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.3332.1 $$\Z/2\Z$$ Not in database
$6$ 6.0.754951232.1 $$\Z/2\Z \times \Z/2\Z$$ 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.