# Properties

 Label 162b1 Conductor 162 Discriminant -648 j-invariant $$-\frac{140625}{8}$$ CM no Rank 0 Torsion Structure $$\Z/{3}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

This curve corresponds to a sporadic degree 3 point on $X_1(21)$. It is the unique elliptic curve over $\Q$ that has a point of order 21 defined over a cubic field [10.4310/MRL.2016.v23.n1.a12, MR:3512885, arXiv:1211.2188].

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -5, 5]); // or

magma: E := EllipticCurve("162b1");

sage: E = EllipticCurve([1, -1, 1, -5, 5]) # or

sage: E = EllipticCurve("162b1")

gp: E = ellinit([1, -1, 1, -5, 5]) \\ or

gp: E = ellinit("162b1")

$$y^2 + x y + y = x^{3} - x^{2} - 5 x + 5$$

## Mordell-Weil group structure

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

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(1, 0\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(1, 0\right)$$, $$\left(1, -2\right)$$

## 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: $$-648$$ = $$-1 \cdot 2^{3} \cdot 3^{4}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{140625}{8}$$ = $$-1 \cdot 2^{-3} \cdot 3^{2} \cdot 5^{6}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$0$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$5.05190741235$$ 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: $$3$$  = $$3\cdot1$$ 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.c

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)

$$q + q^{2} + q^{4} + 2q^{7} + q^{8} - 3q^{11} + 2q^{13} + 2q^{14} + q^{16} - 3q^{17} - q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 6 $$\Gamma_0(N)$$-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])

$$L(E,1)$$ ≈ $$1.68396913745$$

## 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$$(j)_{-}$$
$$2$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$3$$ $$1$$ $$II$$ Additive 1 4 4 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 X4.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 1 & 1 \end{array}\right)$ and has index 2.

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
$$7$$ B.2.1

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 split add ss ordinary 4 - 0,0 0 0 - 0,0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ 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, 7 and 21.
Its isogeny class 162b consists of 4 curves linked by isogenies of degrees dividing 21.

## 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 3.1.648.1 $$\Z/6\Z$$ Not in database
$$\Q(\zeta_{9})^+$$ $$\Z/21\Z$$ 3.3.81.1-8.1-a1
6 6.0.3359232.4 $$\Z/2\Z \times \Z/6\Z$$ Not in database
6.0.177147.2 $$\Z/3\Z \times \Z/3\Z$$ 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.