Properties

 Label 315.b1 Conductor $315$ Discriminant $-9966796875$ j-invariant $$-\frac{250523582464}{13671875}$$ CM no Rank $0$ Torsion structure $$\Z/{3}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, -1182, 16362])

gp: E = ellinit([0, 0, 1, -1182, 16362])

magma: E := EllipticCurve([0, 0, 1, -1182, 16362]);

$$y^2+y=x^3-1182x+16362$$

Mordell-Weil group structure

$\Z/{3}\Z$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(12, 62\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(12, 62\right)$$, $$\left(12, -63\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$315$$ = $3^{2} \cdot 5 \cdot 7$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-9966796875$ = $-1 \cdot 3^{6} \cdot 5^{9} \cdot 7$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{250523582464}{13671875}$$ = $-1 \cdot 2^{15} \cdot 5^{-9} \cdot 7^{-1} \cdot 197^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.67676826796505958000529052010\dots$ Stable Faltings height: $0.12746212363100473430766790164\dots$

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: $1.2730829063823660427179612382\dots$ 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: $9$  = $1\cdot3^{2}\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $3$ 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: $L(E,1)$ ≈ $1.2730829063823660427179612382129197993$

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);

$$q - 2q^{4} + q^{5} + q^{7} + 3q^{11} + 5q^{13} + 4q^{16} - 3q^{17} + 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 180 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$3$ $1$ $I_0^{*}$ Additive -1 2 6 0
$5$ $9$ $I_{9}$ Split multiplicative -1 1 9 9
$7$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

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 $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$3$ 3B.1.1 9.24.0.1

$p$-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 5 7 11 13 17 19 23 29 31 37 41 43 47 ss add split split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 0,5 - 1 1 0 0 0 0 0 0 0 0 0 0 0 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 $d=$ 3 and 9.
Its isogeny class 315.b consists of 3 curves linked by isogenies of degrees dividing 9.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.140.1 $$\Z/6\Z$$ Not in database $3$ 3.3.3969.2 $$\Z/9\Z$$ Not in database $6$ 6.0.686000.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.0.5250987.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.964467.2 $$\Z/9\Z$$ Not in database $9$ 9.3.3501316123704000.11 $$\Z/18\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $18$ 18.0.8549394874383196572862347.2 $$\Z/3\Z \times \Z/9\Z$$ Not in database $18$ 18.0.454044985115170527062208000000.1 $$\Z/3\Z \times \Z/6\Z$$ Not in database $18$ 18.0.137858723094110501552832000000.1 $$\Z/18\Z$$ Not in database $18$ 18.0.10726812773345903701844664000000000.2 $$\Z/2\Z \times \Z/18\Z$$ 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.