# Properties

 Label 131760.u2 Conductor $131760$ Discriminant $-3.138\times 10^{12}$ j-invariant $$-\frac{2081462648832}{28372625}$$ CM no Rank $0$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -12768, -561808])

gp: E = ellinit([0, 0, 0, -12768, -561808])

magma: E := EllipticCurve([0, 0, 0, -12768, -561808]);

## Simplified equation

 $$y^2=x^3-12768x-561808$$ y^2=x^3-12768x-561808 (homogenize, simplify) $$y^2z=x^3-12768xz^2-561808z^3$$ y^2z=x^3-12768xz^2-561808z^3 (dehomogenize, simplify) $$y^2=x^3-12768x-561808$$ y^2=x^3-12768x-561808 (homogenize, minimize)

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$131760$$ = $2^{4} \cdot 3^{3} \cdot 5 \cdot 61$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-3137785344000$ = $-1 \cdot 2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 61^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{2081462648832}{28372625}$$ = $-1 \cdot 2^{15} \cdot 3^{3} \cdot 5^{-3} \cdot 7^{3} \cdot 19^{3} \cdot 61^{-3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.2048006174850847480002896981\dots$ Stable Faltings height: $0.23700036475811201573424626741\dots$

## 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: $0.22438130599383301462122953447\dots$ 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: $3$  = $1\cdot1\cdot1\cdot3$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ 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: $L(E,1)$ ≈ $0.67314391798149904386368860341$

## Modular invariants

Modular form 131760.2.a.u

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 259200 $\Gamma_0(N)$-optimal: yes 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$(j)_{-}$
$2$ $1$ $II^{*}$ Additive -1 4 12 0
$3$ $1$ $II$ Additive -1 3 3 0
$5$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$61$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

## 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
$3$ 3Cs 3.12.0.1

## $p$-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 13 17 19 23 29 31 37 41 43 47 61 add add nonsplit ord ord ord ord ord ord ord ord ord ord ord ord split - - 0 2 0 0 0 0 0 0 0 0 0 2 0 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 $d=$ 3.
Its isogeny class 131760.u consists of 3 curves linked by isogenies of degrees dividing 9.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{3})$$ $$\Z/3\Z$$ Not in database $2$ $$\Q(\sqrt{-1})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.32940.2 $$\Z/2\Z$$ Not in database $4$ $$\Q(\zeta_{12})$$ $$\Z/3\Z \oplus \Z/3\Z$$ Not in database $6$ 6.0.992814894000.2 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $6$ 6.2.52082092800.3 $$\Z/6\Z$$ Not in database $6$ 6.0.17360697600.3 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $18$ 18.6.688441319276820286477612128672563428333248000000000000.1 $$\Z/9\Z$$ Not in database $18$ 18.0.269575566812517305149385143476289536.1 $$\Z/9\Z$$ 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.