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Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, 4, -1]); // or
magma: E := EllipticCurve("162d1");
sage: E = EllipticCurve([1, -1, 1, 4, -1]) # or
sage: E = EllipticCurve("162d1")
gp: E = ellinit([1, -1, 1, 4, -1]) \\ or
gp: E = ellinit("162d1")

$$y^2 + x y + y = x^{3} - x^{2} + 4 x - 1$$

Mordell-Weil group structure

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

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()

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

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

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E) Conductor: $$162$$ = $$2 \cdot 3^{4}$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-5184$$ = $$-1 \cdot 2^{6} \cdot 3^{4}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{109503}{64}$$ = $$2^{-6} \cdot 3^{2} \cdot 23^{3}$$ 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 Real period: $$2.60517025042$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$6$$  = $$( 2 \cdot 3 )\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E) Torsion order: $$3$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form162.2.a.d

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

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

For more coefficients, see the Downloads section to the right.

 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/factorial(ar)

$$L(E,1)$$ ≈ $$1.73678016694$$

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(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

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X20b.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 3 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 1 & 0 \end{array}\right)$ and has index 16.

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]

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 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 $$\Z/12\Z$$ Not in database
6 6.0.177147.2 $$\Z/3\Z \times \Z/3\Z$$ Not in database
6.0.419904.2 $$\Z/4\Z \times \Z/12\Z$$ 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.