# Properties

 Label 6930be1 Conductor $6930$ Discriminant $1.812\times 10^{18}$ j-invariant $$\frac{863913648706111516969}{2486234429521920}$$ 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, -1785767, 916672479])

gp: E = ellinit([1, -1, 1, -1785767, 916672479])

magma: E := EllipticCurve([1, -1, 1, -1785767, 916672479]);

## Simplified equation

 $$y^2+xy+y=x^3-x^2-1785767x+916672479$$ y^2+xy+y=x^3-x^2-1785767x+916672479 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-1785767xz^2+916672479z^3$$ y^2z+xyz+yz^2=x^3-x^2z-1785767xz^2+916672479z^3 (dehomogenize, simplify) $$y^2=x^3-28572267x+58638466406$$ y^2=x^3-28572267x+58638466406 (homogenize, minimize)

## Mordell-Weil group structure

$$\Z/{4}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(419, 15342\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(419, 15342\right)$$, $$\left(419, -15762\right)$$, $$\left(803, -402\right)$$

## 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: $1812464899121479680$ = $2^{28} \cdot 3^{13} \cdot 5 \cdot 7 \cdot 11^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{863913648706111516969}{2486234429521920}$$ = $2^{-28} \cdot 3^{-7} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{-2} \cdot 2593^{3} \cdot 3673^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.3745151590009760760375035597\dots$ Stable Faltings height: $1.8252090146669212303398809412\dots$

## 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: $0.26519408110596137135166523708\dots$ 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: $224$  = $( 2^{2} \cdot 7 )\cdot2^{2}\cdot1\cdot1\cdot2$ 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: $L(E,1)$ ≈ $3.7127171354834591989233133192$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 200704 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $28$ $I_{28}$ Split multiplicative -1 1 28 28
$3$ $4$ $I_{7}^{*}$ Additive -1 2 13 7
$5$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$7$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$11$ $2$ $I_{2}$ Non-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 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 2 - 1 2 0 0 - 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ 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 6930be 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 $K$ $2$ $$\Q(\sqrt{105})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ 4.0.60480.4 $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.4480842240000.3 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.62272557540270000.1 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ 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.