Properties

 Label 277200.hx1 Conductor $277200$ Discriminant $6.421\times 10^{22}$ j-invariant $$\frac{52702650535889104}{22020583921875}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -11159175, 7565103250])

gp: E = ellinit([0, 0, 0, -11159175, 7565103250])

magma: E := EllipticCurve([0, 0, 0, -11159175, 7565103250]);

$$y^2=x^3-11159175x+7565103250$$

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(710, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(710, 0\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$277200$$ = $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $64212022716187500000000$ = $2^{8} \cdot 3^{8} \cdot 5^{12} \cdot 7^{6} \cdot 11^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{52702650535889104}{22020583921875}$$ = $2^{4} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-6} \cdot 11^{-3} \cdot 19^{3} \cdot 41^{3} \cdot 191^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.0734458300971634949681397579\dots$ Stable Faltings height: $1.2573226091727615890253160585\dots$

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: $0.099860800538285170760199189682\dots$ 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: $48$  = $1\cdot2\cdot2^{2}\cdot( 2 \cdot 3 )\cdot1$ 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: $1$ (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: $L(E,1)$ ≈ $1.1983296064594220491223902761858890212$

Modular invariants

Modular form 277200.2.a.hx

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);

$$q + q^{7} - q^{11} - 2q^{13} - 6q^{17} - 8q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 23887872 $\Gamma_0(N)$-optimal: no Manin constant: 1

Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $1$ $I_0^{*}$ Additive -1 4 8 0
$3$ $2$ $I_2^{*}$ Additive -1 2 8 2
$5$ $4$ $I_6^{*}$ Additive 1 2 12 6
$7$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$11$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3

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 2.3.0.1
$3$ 3B 3.4.0.1

$p$-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 $0$.

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 277200.hx consists of 4 curves linked by isogenies of degrees dividing 6.

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 $K$ $2$ $$\Q(\sqrt{11})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{-5})$$ $$\Z/6\Z$$ Not in database $4$ 4.0.1940400.1 $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-5}, \sqrt{11})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.2.5668704000.1 $$\Z/6\Z$$ Not in database $8$ 8.0.7289334581760000.117 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.60242434560000.40 $$\Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $18$ 18.0.4579270566836444727686387387015206261248000000000000000.1 $$\Z/18\Z$$ 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.