# Properties

 Label 162.d2 Conductor $162$ Discriminant $-5184$ j-invariant $$\frac{109503}{64}$$ CM no Rank $0$ Torsion structure $$\Z/{3}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-x^2+4x-1$$ y^2+xy+y=x^3-x^2+4x-1 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z+4xz^2-z^3$$ y^2z+xyz+yz^2=x^3-x^2z+4xz^2-z^3 (dehomogenize, simplify) $$y^2=x^3+69x+22$$ y^2=x^3+69x+22 (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 1, 4, -1])

gp: E = ellinit([1, -1, 1, 4, -1])

magma: E := EllipticCurve([1, -1, 1, 4, -1]);

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);

$$\left(1, 1\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(1, 1\right)$$, $$\left(1, -3\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$162$$ = $2 \cdot 3^{4}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-5184$ = $-1 \cdot 2^{6} \cdot 3^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{109503}{64}$$ = $2^{-6} \cdot 3^{2} \cdot 23^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.59780648594258934269379073217\dots$ Stable Faltings height: $-0.96401058216529257315887247781\dots$

## 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: $2.6051702504152866173475482491\dots$ 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: $6$  = $( 2 \cdot 3 )\cdot1$ 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: $L(E,1)$ ≈ $1.7367801669435244115650321660$

## 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);

$$q + q^{2} + q^{4} + 3 q^{5} - 4 q^{7} + q^{8} + 3 q^{10} - q^{13} - 4 q^{14} + q^{16} + 3 q^{17} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 12 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$3$ $1$ $II$ Additive 1 4 4 0

## 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
$2$ 2G 4.16.0.2
$3$ 3B.1.1 3.8.0.1
sage: gens = [[9, 4, 8, 5], [7, 3, 0, 1], [9, 4, 4, 9], [1, 9, 9, 10]]

sage: GL(2,Integers(12)).subgroup(gens)

magma: Gens := [[9, 4, 8, 5], [7, 3, 0, 1], [9, 4, 4, 9], [1, 9, 9, 10]];

magma: sub<GL(2,Integers(12))|Gens>;

The image of the adelic Galois representation has level $12$, index $128$, genus $1$, and generators

$\left(\begin{array}{rr} 9 & 4 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 7 & 3 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 4 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 10 \end{array}\right)$

## $p$-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 2 3 split add 4 - 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 162.d 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 $K$ $3$ 3.1.324.1 $$\Z/12\Z$$ Not in database $6$ 6.0.419904.2 $$\Z/4\Z \oplus \Z/12\Z$$ Not in database $6$ 6.0.177147.2 $$\Z/3\Z \oplus \Z/3\Z$$ Not in database $9$ 9.3.74384733888.1 $$\Z/9\Z$$ Not in database $12$ 12.2.722204136308736.4 $$\Z/24\Z$$ Not in database $18$ 18.0.16599265906765726789632.5 $$\Z/3\Z \oplus \Z/12\Z$$ Not in database

We only show fields where the torsion growth is primitive.