Properties

Label 162.a2
Conductor 162
Discriminant -3779136
j-invariant \( \frac{109503}{64} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, 39, -19]); // or
magma: E := EllipticCurve("162a2");
sage: E = EllipticCurve([1, -1, 0, 39, -19]) # or
sage: E = EllipticCurve("162a2")
gp: E = ellinit([1, -1, 0, 39, -19]) \\ or
gp: E = ellinit("162a2")

\( y^2 + x y = x^{3} - x^{2} + 39 x - 19 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(10, -41\right) \)
\(\hat{h}(P)\) ≈  0.101978294623

Integral points

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

\( \left(1, 4\right) \), \( \left(2, 7\right) \), \( \left(10, 31\right) \), \( \left(19, 76\right) \), \( \left(154, 1831\right) \)

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: \(-3779136 \)  =  \(-1 \cdot 2^{6} \cdot 3^{10} \)
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: \(1\)
magma: Regulator(E);
sage: E.regulator()
Regulator: \(0.101978294623\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(1.46514758762\)
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: \(1\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 162.2.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)

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

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 36
\( \Gamma_0(N) \)-optimal: no
Manin constant: not computed

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\((j)_{-}\)
\(2\) \(2\) \( I_{6} \) Non-split multiplicative 1 1 6 6
\(3\) \(3\) \( IV^{*} \) Additive 1 4 10 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.1.2

$p$-adic data

$p$-adic regulators

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

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit add ordinary ordinary ss ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 4 - 1 1 1,1 3 1 1 1,1 1 1 1 1 1 3
$\mu$-invariant(s) 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 162.a 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{-3}) \) \(\Z/3\Z\) 2.0.3.1-2916.1-a2
3 3.1.324.1 \(\Z/2\Z\) Not in database
3.1.243.1 \(\Z/3\Z\) Not in database
6 6.2.1259712.2 \(\Z/4\Z\) Not in database
6.0.419904.2 \(\Z/2\Z \times \Z/2\Z\) Not in database
6.0.314928.2 \(\Z/12\Z\) 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.