# Properties

 Label 6720bi2 Conductor $6720$ Discriminant $3.750\times 10^{17}$ j-invariant $$\frac{448487713888272974160064}{91549016015625}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-x^2-25515001x-49598319815$$ y^2=x^3-x^2-25515001x-49598319815 (homogenize, simplify) $$y^2z=x^3-x^2z-25515001xz^2-49598319815z^3$$ y^2z=x^3-x^2z-25515001xz^2-49598319815z^3 (dehomogenize, simplify) $$y^2=x^3-2066715108x-36163375290432$$ y^2=x^3-2066715108x-36163375290432 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, -25515001, -49598319815])

gp: E = ellinit([0, -1, 0, -25515001, -49598319815])

magma: E := EllipticCurve([0, -1, 0, -25515001, -49598319815]);

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);

$$\left(-2915, 0\right)$$, $$\left(5833, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-2917, 0\right)$$, $$\left(-2915, 0\right)$$, $$\left(5833, 0\right)$$

## 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: $374984769600000000$ = $2^{12} \cdot 3^{14} \cdot 5^{8} \cdot 7^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{448487713888272974160064}{91549016015625}$$ = $2^{6} \cdot 3^{-14} \cdot 5^{-8} \cdot 7^{-2} \cdot 19136251^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.7599875578878890171728697676\dots$ Stable Faltings height: $2.0668403773279437077556376461\dots$

## 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: $0.067173915118157601347276711424\dots$ 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: $32$  = $2^{2}\cdot2\cdot2\cdot2$ 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: $9$ = $3^2$ (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: $L(E,1)$ ≈ $1.2091304721268368242509808056$

## 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);

$$q - q^{3} - q^{5} - q^{7} + q^{9} - 4 q^{11} + 6 q^{13} + q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 430080 $\Gamma_0(N)$-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
$3$ $2$ $I_{14}$ Non-split multiplicative 1 1 14 14
$5$ $2$ $I_{8}$ Non-split multiplicative 1 1 8 8
$7$ $2$ $I_{2}$ 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
$2$ 2Cs 8.12.0.1
sage: gens = [[45, 4, 8, 11], [73, 2, 0, 1], [1, 4, 0, 1], [1, 0, 4, 1], [3, 2, 110, 167], [81, 166, 2, 1], [165, 4, 164, 5]]

sage: GL(2,Integers(168)).subgroup(gens)

magma: Gens := [[45, 4, 8, 11], [73, 2, 0, 1], [1, 4, 0, 1], [1, 0, 4, 1], [3, 2, 110, 167], [81, 166, 2, 1], [165, 4, 164, 5]];

magma: sub<GL(2,Integers(168))|Gens>;

The image of the adelic Galois representation has level $168$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 45 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 73 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 110 & 167 \end{array}\right),\left(\begin{array}{rr} 81 & 166 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 165 & 4 \\ 164 & 5 \end{array}\right)$

## $p$-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 nonsplit nonsplit - 2 0 0 - 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 6720bi 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 $K$ $4$ $$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{14})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{7})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ deg 8 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ 16.0.162447943996702457856.1 $$\Z/4\Z \oplus \Z/4\Z$$ 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 $$\Z/2\Z \oplus \Z/8\Z$$ 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.