Properties

Label 22696a1
Conductor 22696
Discriminant 726272
j-invariant \( \frac{504871936}{2837} \)
CM no
Rank 3
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -105, 379]); // or
magma: E := EllipticCurve("22696a1");
sage: E = EllipticCurve([0, 1, 0, -105, 379]) # or
sage: E = EllipticCurve("22696a1")
gp: E = ellinit([0, 1, 0, -105, 379]) \\ or
gp: E = ellinit("22696a1")

\( y^2 = x^{3} + x^{2} - 105 x + 379 \)

Mordell-Weil group structure

\(\Z^3\)

Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(45, -298\right) \)\( \left(-9, 26\right) \)\( \left(-3, 26\right) \)
\(\hat{h}(P)\) ≈  3.111897939181.881000291231.72799047686

Integral points

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

\( \left(-11, 18\right) \), \( \left(-10, 23\right) \), \( \left(-9, 26\right) \), \( \left(-3, 26\right) \), \( \left(-1, 22\right) \), \( \left(3, 10\right) \), \( \left(5, 2\right) \), \( \left(6, 1\right) \), \( \left(7, 6\right) \), \( \left(11, 26\right) \), \( \left(14, 43\right) \), \( \left(27, 134\right) \), \( \left(35, 202\right) \), \( \left(45, 298\right) \), \( \left(61, 474\right) \), \( \left(503, 11290\right) \), \( \left(579, 13942\right) \), \( \left(1701, 70174\right) \), \( \left(2419, 118998\right) \), \( \left(109141, 36056526\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 22696 \)  =  \(2^{3} \cdot 2837\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(726272 \)  =  \(2^{8} \cdot 2837 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{504871936}{2837} \)  =  \(2^{10} \cdot 79^{3} \cdot 2837^{-1}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
Rank: \(3\)
magma: Regulator(E);
sage: E.regulator()
Regulator: \(0.48300720723\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(2.8673192008\)
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: \( 4 \)  = \( 2^{2}\cdot1 \)
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: \(1\) (rounded)

Modular invariants

Modular form 22696.2.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)

\( q - 2q^{3} - 4q^{5} - 3q^{7} + q^{9} - 6q^{11} - 6q^{13} + 8q^{15} - 3q^{17} - 5q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 8768
\( \Gamma_0(N) \)-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])

\( L^{(3)}(E,1)/3! \) ≈ \( 5.53974335766 \)

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\((j)_{-}\)
\(2\) \(4\) \( I_1^{*} \) Additive 1 3 8 0
\(2837\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 \( p \) .

$p$-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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 2837
Reduction type add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary nonsplit
$\lambda$-invariant(s) - 3 3 3 3 3 3 3 3 3 3 3 3 3,3 3 ?
$\mu$-invariant(s) - 0 0 0 0 0 0 0 0 0 0 0 0 0,0 0 ?

An entry ? indicates that the invariants have not yet been computed.

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has no rational isogenies. Its isogeny class 22696a consists of this curve only.

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
3 3.3.11348.1 \(\Z/2\Z\) Not in database
6 6.6.365340644048.1 \(\Z/2\Z \times \Z/2\Z\) 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.