# Properties

 Label 162a1 Conductor 162 Discriminant -2916 j-invariant $$-\frac{35937}{4}$$ CM no Rank 1 Torsion Structure $$\Z/{3}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -6, 8]); // or
magma: E := EllipticCurve("162a1");
sage: E = EllipticCurve([1, -1, 0, -6, 8]) # or
sage: E = EllipticCurve("162a1")
gp: E = ellinit([1, -1, 0, -6, 8]) \\ or
gp: E = ellinit("162a1")

$$y^2 + x y = x^{3} - x^{2} - 6 x + 8$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-2, 4\right)$$ $$\hat{h}(P)$$ ≈ 0.305934883869

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

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

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$$\left(-2, 4\right)$$, $$\left(-1, 4\right)$$, $$\left(1, 1\right)$$, $$\left(2, 0\right)$$, $$\left(4, 4\right)$$, $$\left(7, 13\right)$$, $$\left(46, 286\right)$$

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$162$$ = $$2 \cdot 3^{4}$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-2916$$ = $$-1 \cdot 2^{2} \cdot 3^{6}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$-\frac{35937}{4}$$ = $$-1 \cdot 2^{-2} \cdot 3^{3} \cdot 11^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$1$$ magma: Regulator(E); sage: E.regulator() Regulator: $$0.305934883869$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$4.39544276285$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$6$$  = $$2\cdot3$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$3$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form162.2.a.a

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 12 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])

$$L'(E,1)$$ ≈ $$0.896479514136$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$3$$ $$3$$ $$IV$$ Additive 1 4 6 0

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X20.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 3 & 0 \end{array}\right)$ and has index 8.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$3$$ B.1.1

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ of good ordinary reduction.

## 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 nonsplit add ordinary ordinary ss ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary 4 - 1 1 1,1 3 1 1 1,1 1 1 1 1 1 3 0 - 0 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 162a consists of 2 curves linked by isogenies of degree 3.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.324.1 $$\Z/6\Z$$ Not in database
6 6.2.1259712.2 $$\Z/12\Z$$ Not in database
6.0.419904.2 $$\Z/2\Z \times \Z/6\Z$$ Not in database
6.0.314928.2 $$\Z/12\Z$$ Not in database
6.0.34992.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.