# Properties

 Label 378b3 Conductor $378$ Discriminant $-3402$ j-invariant $$-\frac{545407363875}{14}$$ CM no Rank $0$ Torsion structure $$\Z/{3}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy=x^3-x^2-1062x+13590$$ y^2+xy=x^3-x^2-1062x+13590 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z-1062xz^2+13590z^3$$ y^2z+xyz=x^3-x^2z-1062xz^2+13590z^3 (dehomogenize, simplify) $$y^2=x^3-16995x+852766$$ y^2=x^3-16995x+852766 (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 0, -1062, 13590])

gp: E = ellinit([1, -1, 0, -1062, 13590])

magma: E := EllipticCurve([1, -1, 0, -1062, 13590]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z/{3}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(19, -9\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(19, -9\right)$$, $$\left(19, -10\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$378$$ = $2 \cdot 3^{3} \cdot 7$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-3402$ = $-1 \cdot 2 \cdot 3^{5} \cdot 7$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{545407363875}{14}$$ = $-1 \cdot 2^{-1} \cdot 3 \cdot 5^{3} \cdot 7^{-1} \cdot 11^{3} \cdot 103^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.19388385769632813006906179683\dots$ Stable Faltings height: $-0.26387126258205090801229038522\dots$

## 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: $3.2456774249956641238240628560\dots$ 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: $3$  = $1\cdot3\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $3$ 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: $L(E,1)$ ≈ $1.0818924749985547079413542853$

## 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);

$$q - q^{2} + q^{4} + q^{7} - q^{8} + 5 q^{13} - q^{14} + q^{16} - 3 q^{17} + 2 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 108 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 3 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
$3$ $3$ $IV$ Additive 1 3 5 0
$7$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

## 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
$3$ 3B.1.1 9.24.0.1

## $p$-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) 2 3 7 nonsplit add split 1 - 1 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 $d=$ 3 and 9.
Its isogeny class 378b 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.1512.1 $$\Z/6\Z$$ Not in database $3$ $$\Q(\zeta_{7})^+$$ $$\Z/9\Z$$ Not in database $6$ 6.0.384072192.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $6$ 6.0.6805279152.2 $$\Z/3\Z \oplus \Z/3\Z$$ Not in database $6$ 6.0.138883248.16 $$\Z/9\Z$$ Not in database $9$ 9.3.8299415996928.1 $$\Z/18\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $18$ 18.0.315164892649262158461997559808.4 $$\Z/3\Z \oplus \Z/9\Z$$ Not in database $18$ 18.0.20494572895025913969633989670960365568.2 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database $18$ 18.0.8535848769273600153950016522682368.1 $$\Z/18\Z$$ Not in database $18$ 18.0.6665409440369750708186945421312.1 $$\Z/2\Z \oplus \Z/18\Z$$ 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.