Properties

 Label 6930.bb4 Conductor $6930$ Discriminant $-2.182\times 10^{12}$ j-invariant $$\frac{1865864036231}{2993760000}$$ CM no Rank $0$ Torsion structure $$\Z/{4}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, 2308, -57409])

gp: E = ellinit([1, -1, 1, 2308, -57409])

magma: E := EllipticCurve([1, -1, 1, 2308, -57409]);

$$y^2+xy+y=x^3-x^2+2308x-57409$$

Mordell-Weil group structure

$\Z/{4}\Z$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(81, 769\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(21, -11\right)$$, $$\left(81, 769\right)$$, $$\left(81, -851\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$6930$$ = $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-2182451040000$ = $-1 \cdot 2^{8} \cdot 3^{11} \cdot 5^{4} \cdot 7 \cdot 11$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1865864036231}{2993760000}$$ = $2^{-8} \cdot 3^{-5} \cdot 5^{-4} \cdot 7^{-1} \cdot 11^{-1} \cdot 13^{3} \cdot 947^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.0527855842408225246649517441\dots$ Stable Faltings height: $0.50347943990676767896732912564\dots$

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: $0.43395657998669944583037065659\dots$ 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: $128$  = $2^{3}\cdot2^{2}\cdot2^{2}\cdot1\cdot1$ 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) 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: $L(E,1)$ ≈ $3.4716526398935955666429652528$

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);

$$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} - q^{11} - 2 q^{13} - q^{14} + q^{16} + 2 q^{17} + 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 10240 $\Gamma_0(N)$-optimal: yes 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$(j)_{-}$
$2$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$3$ $4$ $I_{5}^{*}$ Additive -1 2 11 5
$5$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$7$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$11$ $1$ $I_{1}$ Non-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 for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 4.12.0.7

$p$-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 split add split nonsplit nonsplit 4 - 1 0 0 0 - 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 and 4.
Its isogeny class 6930.bb consists of 4 curves linked by isogenies of degrees dividing 4.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-231})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ 4.2.831600.7 $$\Z/8\Z$$ Not in database $8$ 8.0.38896618261729536.14 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ 8.0.4100250702240000.8 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.0.10551382992005625.7 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.2.62272557540270000.13 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ 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.