# Properties

 Label 6930.bj2 Conductor $6930$ Discriminant $-3.922\times 10^{14}$ j-invariant $$-\frac{28124139978713043}{14526050000000}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -19007, 1392231])

gp: E = ellinit([1, -1, 1, -19007, 1392231])

magma: E := EllipticCurve([1, -1, 1, -19007, 1392231]);

$$y^2+xy+y=x^3-x^2-19007x+1392231$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-69, 1574\right)$$ (-69, 1574) $\hat{h}(P)$ ≈ $0.058520269525866281653826050524$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{661}{4}, \frac{657}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-139, 1224\right)$$, $$\left(-139, -1086\right)$$, $$\left(-69, 1574\right)$$, $$\left(-69, -1506\right)$$, $$\left(-9, 1254\right)$$, $$\left(-9, -1246\right)$$, $$\left(41, 804\right)$$, $$\left(41, -846\right)$$, $$\left(71, 594\right)$$, $$\left(71, -666\right)$$, $$\left(75, 582\right)$$, $$\left(75, -658\right)$$, $$\left(141, 1154\right)$$, $$\left(141, -1296\right)$$, $$\left(239, 3114\right)$$, $$\left(239, -3354\right)$$, $$\left(491, 10254\right)$$, $$\left(491, -10746\right)$$, $$\left(1691, 68454\right)$$, $$\left(1691, -70146\right)$$, $$\left(2241, 104754\right)$$, $$\left(2241, -106996\right)$$, $$\left(97511, 30400614\right)$$, $$\left(97511, -30498126\right)$$

## 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: $-392203350000000$ = $-1 \cdot 2^{7} \cdot 3^{3} \cdot 5^{8} \cdot 7^{4} \cdot 11^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{28124139978713043}{14526050000000}$$ = $-1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{-8} \cdot 7^{-4} \cdot 11^{-2} \cdot 167^{3} \cdot 607^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.5047582364612556239146649746\dots$ Stable Faltings height: $1.2301051642942282010658536654\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.058520269525866281653826050524\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.49695301489799535364135058471\dots$ 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: $896$  = $7\cdot2\cdot2^{3}\cdot2^{2}\cdot2$ 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: $1$ (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: $L'(E,1)$ ≈ $6.5143286596690466655476616383$

## 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);

$$q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} + q^{11} - 4 q^{13} + q^{14} + q^{16} - 2 q^{17} - 6 q^{19} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 28672 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $7$ $I_{7}$ Split multiplicative -1 1 7 7
$3$ $2$ $III$ Additive 1 2 3 0
$5$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$7$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$11$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

## 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

## $p$-adic regulators

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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

## 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 split add split split split ord ord ord ord ord ss ord ord ord ord 4 - 2 2 2 1 1 1 1 1 1,3 1 1 1 1 1 - 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 $d=$ 2.
Its isogeny class 6930.bj consists of 2 curves linked by isogenies of degree 2.

## 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 $K$ $2$ $$\Q(\sqrt{-6})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.2.104544.1 $$\Z/4\Z$$ Not in database $8$ 8.0.369975361536.15 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.699484667904.4 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.2.768797006670000.9 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ 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.