# Properties

 Label 5054c1 Conductor $5054$ Discriminant $-1317284668$ j-invariant $$-\frac{15625}{28}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3+x^2-188x-2087$$ y^2+xy+y=x^3+x^2-188x-2087 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3+x^2z-188xz^2-2087z^3$$ y^2z+xyz+yz^2=x^3+x^2z-188xz^2-2087z^3 (dehomogenize, simplify) $$y^2=x^3-243675x-93707658$$ y^2=x^3-243675x-93707658 (homogenize, minimize)

sage: E = EllipticCurve([1, 1, 1, -188, -2087])

gp: E = ellinit([1, 1, 1, -188, -2087])

magma: E := EllipticCurve([1, 1, 1, -188, -2087]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(17, -9\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(17, -9\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$5054$$ = $2 \cdot 7 \cdot 19^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-1317284668$ = $-1 \cdot 2^{2} \cdot 7 \cdot 19^{6}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{15625}{28}$$ = $-1 \cdot 2^{-2} \cdot 5^{6} \cdot 7^{-1}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.44013459105822234148143477313\dots$ Stable Faltings height: $-1.0320848985249978885230789428\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.60817709098151674228887984272\dots$ 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: $4$  = $2\cdot1\cdot2$ 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: $9$ = $3^2$ (exact) 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); Special value: $L(E,1)$ ≈ $5.4735938188336506805999185845$

## Modular invariants

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2376 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

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$(j)_{-}$
$2$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$7$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$19$ $2$ $I_0^{*}$ Additive -1 2 6 0

## Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.6.0.1
$3$ 3B 9.12.0.1

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $p$-adic regulators are identically $1$ since the rank is $0$.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 7 19 split ord split add 2 2 1 - 0 0 0 -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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, 3, 6, 9 and 18.
Its isogeny class 5054c consists of 6 curves linked by isogenies of degrees dividing 18.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-7})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{-19})$$ $$\Z/6\Z$$ Not in database $4$ 4.2.161728.1 $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-7}, \sqrt{-19})$$ $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $6$ 6.2.7114374288.1 $$\Z/6\Z$$ Not in database $6$ 6.0.16468459.1 $$\Z/18\Z$$ Not in database $8$ 8.0.15700106576896.7 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.1281641353216.14 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.26155945984.3 $$\Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/18\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $12$ 12.0.13289296949899369.1 $$\Z/2\Z \oplus \Z/18\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.