# Properties

 Label 405a1 Conductor $405$ Discriminant $10125$ j-invariant $$\frac{2359296}{125}$$ CM no Rank $0$ Torsion structure $$\Z/{3}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+y=x^3-12x+15$$ y^2+y=x^3-12x+15 (homogenize, simplify) $$y^2z+yz^2=x^3-12xz^2+15z^3$$ y^2z+yz^2=x^3-12xz^2+15z^3 (dehomogenize, simplify) $$y^2=x^3-192x+976$$ y^2=x^3-192x+976 (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 1, -12, 15])

gp: E = ellinit([0, 0, 1, -12, 15])

magma: E := EllipticCurve([0, 0, 1, -12, 15]);

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);

$$\left(3, 2\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(3, 2\right)$$, $$\left(3, -3\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$405$$ = $3^{4} \cdot 5$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $10125$ = $3^{4} \cdot 5^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2359296}{125}$$ = $2^{18} \cdot 3^{2} \cdot 5^{-3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.47463585994036034429740154432\dots$ Stable Faltings height: $-0.84083995616306357476248328996\dots$

## 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: $4.0159643400787201897604015173\dots$ 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: $3$  = $1\cdot3$ 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: $L(E,1)$ ≈ $1.3386547800262400632534671724$

## 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);

$$q - 2 q^{4} + q^{5} + 2 q^{7} + 3 q^{11} - 4 q^{13} + 4 q^{16} + 6 q^{17} - q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 24 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 2 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$(j)_{-}$
$3$ $1$ $II$ Additive 1 4 4 0
$5$ $3$ $I_{3}$ 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
$3$ 3B.1.1 3.8.0.1

The image of the adelic Galois representation has level $30$, index $16$, and genus $0$.

## $p$-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 5 ss add split 0,3 - 1 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 $d=$ 3.
Its isogeny class 405a consists of 2 curves linked by isogenies of degree 3.

## 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 $K$ $3$ 3.3.1620.1 $$\Z/6\Z$$ 3.3.1620.1-25.1-a2 $6$ 6.6.13122000.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $6$ 6.0.177147.2 $$\Z/3\Z \oplus \Z/3\Z$$ Not in database $9$ 9.3.18160335421875.2 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $18$ 18.0.259363529793214481088000000.1 $$\Z/3\Z \oplus \Z/6\Z$$ 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.