# Properties

 Label 6720.bg2 Conductor $6720$ Discriminant $1.452\times 10^{13}$ j-invariant $$\frac{13507798771700416}{3544416225}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

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Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-x^2-79385x+8633625$$ y^2=x^3-x^2-79385x+8633625 (homogenize, simplify) $$y^2z=x^3-x^2z-79385xz^2+8633625z^3$$ y^2z=x^3-x^2z-79385xz^2+8633625z^3 (dehomogenize, simplify) $$y^2=x^3-6430212x+6274622016$$ y^2=x^3-6430212x+6274622016 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, -79385, 8633625])

gp: E = ellinit([0, -1, 0, -79385, 8633625])

magma: E := EllipticCurve([0, -1, 0, -79385, 8633625]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z/{2}\Z \oplus \Z/{2}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(161, 0\right)$$, $$\left(165, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-325, 0\right)$$, $$\left(161, 0\right)$$, $$\left(165, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$6720$$ = $2^{6} \cdot 3 \cdot 5 \cdot 7$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $14517928857600$ = $2^{12} \cdot 3^{10} \cdot 5^{2} \cdot 7^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{13507798771700416}{3544416225}$$ = $2^{6} \cdot 3^{-10} \cdot 5^{-2} \cdot 7^{-4} \cdot 59539^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.5102570373606597235040814407\dots$ Stable Faltings height: $0.81710985680071441408684931924\dots$

## 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: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $0.68595570283831005637099973551\dots$ 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: $64$  = $2^{2}\cdot2\cdot2\cdot2^{2}$ 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: $L(E,1)$ ≈ $2.7438228113532402254839989421$

## 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);

$$q - q^{3} + q^{5} + q^{7} + q^{9} + 4 q^{11} + 6 q^{13} - q^{15} + 6 q^{17} + O(q^{20})$$

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 30720 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 4 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$(j)_{-}$
$2$ $4$ $I_{2}^{*}$ Additive -1 6 12 0
$3$ $2$ $I_{10}$ Non-split multiplicative 1 1 10 10
$5$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$7$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

## 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
$2$ 2Cs 4.12.0.1
sage: gens = [[41, 4, 82, 9], [57, 118, 2, 1], [117, 4, 116, 5], [27, 116, 112, 109], [97, 2, 0, 1], [1, 4, 0, 1], [1, 0, 4, 1]]

sage: GL(2,Integers(120)).subgroup(gens)

magma: Gens := [[41, 4, 82, 9], [57, 118, 2, 1], [117, 4, 116, 5], [27, 116, 112, 109], [97, 2, 0, 1], [1, 4, 0, 1], [1, 0, 4, 1]];

magma: sub<GL(2,Integers(120))|Gens>;

The image of the adelic Galois representation has level $120$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 41 & 4 \\ 82 & 9 \end{array}\right),\left(\begin{array}{rr} 57 & 118 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 117 & 4 \\ 116 & 5 \end{array}\right),\left(\begin{array}{rr} 27 & 116 \\ 112 & 109 \end{array}\right),\left(\begin{array}{rr} 97 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$

## $p$-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 add nonsplit split split - 0 1 1 - 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.
Its isogeny class 6720.bg 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{10})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-6}, \sqrt{-10})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{6})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.3317760000.5 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.4.22658678784000000.31 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ deg 8 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ 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.