# Properties

 Label 177870.bs5 Conductor $177870$ Discriminant $1.191\times 10^{19}$ j-invariant $$\frac{5203798902289}{57153600}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -2140492, 1192980496]) # or

sage: E = EllipticCurve("177870hp2")

gp: E = ellinit([1, 1, 0, -2140492, 1192980496]) \\ or

gp: E = ellinit("177870hp2")

magma: E := EllipticCurve([1, 1, 0, -2140492, 1192980496]); // or

magma: E := EllipticCurve("177870hp2");

$$y^2 + x y = x^{3} + x^{2} - 2140492 x + 1192980496$$

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(776, -388\right)$$, $$\left(\frac{3643}{4}, -\frac{3643}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1688, 844\right)$$, $$\left(776, -388\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$177870$$ = $$2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$11912089342654670400$$ = $$2^{6} \cdot 3^{6} \cdot 5^{2} \cdot 7^{8} \cdot 11^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{5203798902289}{57153600}$$ = $$2^{-6} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{-2} \cdot 13^{3} \cdot 31^{3} \cdot 43^{3}$$ 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: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.226829964486$$ 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: $$128$$  = $$2\cdot2\cdot2\cdot2^{2}\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

Modular form 177870.2.a.bs

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 6635520 $$\Gamma_0(N)$$-optimal: no 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);

$$L(E,1)$$ ≈ $$1.81463971589$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$3$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$5$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$7$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$11$$ $$4$$ $$I_0^{*}$$ Additive -1 2 6 0

## 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.

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
$$2$$ Cs
$$3$$ B

## $p$-adic data

### $p$-adic regulators

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

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

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 177870.bs consists of 8 curves linked by isogenies of degrees dividing 12.

## 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
$2$ $$\Q(\sqrt{77})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
$4$ $$\Q(\sqrt{-11}, \sqrt{-14})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$4$ $$\Q(\sqrt{11}, \sqrt{105})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$4$ $$\Q(\sqrt{30}, \sqrt{-154})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$6$ 6.0.377495724375.1 $$\Z/2\Z \times \Z/6\Z$$ 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.