Properties

Label 363726j1
Conductor 363726
Discriminant -124228886874048
j-invariant \( -\frac{10218313}{96192} \)
CM no
Rank 0
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -4923, -551259]); // or
magma: E := EllipticCurve("363726j1");
sage: E = EllipticCurve([1, -1, 0, -4923, -551259]) # or
sage: E = EllipticCurve("363726j1")
gp: E = ellinit([1, -1, 0, -4923, -551259]) \\ or
gp: E = ellinit("363726j1")

\( y^2 + x y = x^{3} - x^{2} - 4923 x - 551259 \)

Mordell-Weil group structure

\(\Z/{2}\Z\)

Torsion generators

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

\( \left(102, -51\right) \)

Integral points

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

\( \left(102, -51\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 363726 \)  =  \(2 \cdot 3^{2} \cdot 11^{2} \cdot 167\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(-124228886874048 \)  =  \(-1 \cdot 2^{6} \cdot 3^{8} \cdot 11^{6} \cdot 167 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( -\frac{10218313}{96192} \)  =  \(-1 \cdot 2^{-6} \cdot 3^{-2} \cdot 7^{3} \cdot 31^{3} \cdot 167^{-1}\)
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: \(0.248836409017\)
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: \( 32 \)  = \( 2\cdot2^{2}\cdot2^{2}\cdot1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(2\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 363726.2.a.j

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)

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

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 1474560
\( \Gamma_0(N) \)-optimal: unknown* (one of 2 curves in this isogeny class which might be optimal)
Manin constant: 1 (conditional*)
* The optimal curve in each isogeny class has not been determined in all cases for conductors over 270000. The Manin constant is correct provided that this curve is optimal.

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])

\( L(E,1) \) ≈ \( 1.99069127214 \)

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
\(3\) \(4\) \( I_2^{*} \) Additive -1 2 8 2
\(11\) \(4\) \( I_0^{*} \) Additive -1 2 6 0
\(167\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1

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 X6.

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

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
\(2\) B

$p$-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 \(0\).

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 363726j consists of 2 curves linked by isogenies of degree 2.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{-167}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
4 \(x^{4} \) \(\mathstrut -\mathstrut 2 x^{3} \) \(\mathstrut +\mathstrut 37 x^{2} \) \(\mathstrut +\mathstrut 162 x \) \(\mathstrut -\mathstrut 1293 \) \(\Z/4\Z\) 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.