# Properties

 Label 6930q4 Conductor $6930$ Discriminant $-5.933\times 10^{15}$ j-invariant $$-\frac{1573398910560073969}{8138108343750}$$ CM no Rank $1$ Torsion structure $$\Z/{6}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -218079, 39427803])

gp: E = ellinit([1, -1, 0, -218079, 39427803])

magma: E := EllipticCurve([1, -1, 0, -218079, 39427803]);

$$y^2+xy=x^3-x^2-218079x+39427803$$

## Mordell-Weil group structure

$\Z\times \Z/{6}\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(459, 5760\right)$$ (459, 5760) $\hat{h}(P)$ ≈ $0.98619519273464716296585744175$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(327, 1569\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-333, 8829\right)$$, $$\left(-333, -8496\right)$$, $$\left(117, 3879\right)$$, $$\left(117, -3996\right)$$, $$\left(217, 1404\right)$$, $$\left(217, -1621\right)$$, $$\left(273, 291\right)$$, $$\left(273, -564\right)$$, $$\left(327, 1569\right)$$, $$\left(327, -1896\right)$$, $$\left(459, 5760\right)$$, $$\left(459, -6219\right)$$, $$\left(2637, 132084\right)$$, $$\left(2637, -134721\right)$$

## 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: $-5932680982593750$ = $-1 \cdot 2 \cdot 3^{7} \cdot 5^{6} \cdot 7^{2} \cdot 11^{6}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{1573398910560073969}{8138108343750}$$ = $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 5^{-6} \cdot 7^{-2} \cdot 11^{-6} \cdot 17^{3} \cdot 31^{3} \cdot 2207^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.8725899355142528671910083475\dots$ Stable Faltings height: $1.3232837911801980214933857290\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.98619519273464716296585744175\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.42799370292970511463243075538\dots$ 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: $288$  = $1\cdot2^{2}\cdot( 2 \cdot 3 )\cdot2\cdot( 2 \cdot 3 )$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $6$ 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: $L'(E,1)$ ≈ $3.3766826587998068612229752735$

## 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);

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

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 82944 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$3$ $4$ $I_{1}^{*}$ Additive -1 2 7 1
$5$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$7$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$11$ $6$ $I_{6}$ Split multiplicative -1 1 6 6

## 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.1.1 3.8.0.1

## $p$-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 nonsplit add split split split ord ss ord ord ord ord ord ord ord ord 2 - 4 2 2 3 1,1 1 1 3 3 1 3 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, 3 and 6.
Its isogeny class 6930q 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-6})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $4$ 4.2.14229600.7 $$\Z/12\Z$$ Not in database $6$ 6.0.6805279152.1 $$\Z/3\Z \times \Z/6\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/12\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/12\Z$$ Not in database $9$ 9.3.15258333411304656750000.3 $$\Z/18\Z$$ Not in database $12$ Deg 12 $$\Z/6\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/24\Z$$ Not in database $18$ 18.0.5859025065106854728634977947901171859456000000000000.1 $$\Z/2\Z \times \Z/18\Z$$ 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.