Minimal Weierstrass equation
magma: E := EllipticCurve([0, 0, 0, 292, 32]); // or
magma: E := EllipticCurve("498080f1");
sage: E = EllipticCurve([0, 0, 0, 292, 32]) # or
sage: E = EllipticCurve("498080f1")
gp: E = ellinit([0, 0, 0, 292, 32]) \\ or
gp: E = ellinit("498080f1")
\( y^2 = x^{3} + 292 x + 32 \)
Mordell-Weil group structure
Trivial
Integral points
magma: IntegralPoints(E);
sage: E.integral_points()
None
Invariants
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magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 498080 \) | = | \(2^{5} \cdot 5 \cdot 11 \cdot 283\) | ||
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magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(-1593856000 \) | = | \(-1 \cdot 2^{12} \cdot 5^{3} \cdot 11 \cdot 283 \) | ||
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magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( \frac{672221376}{389125} \) | = | \(2^{6} \cdot 3^{3} \cdot 5^{-3} \cdot 11^{-1} \cdot 73^{3} \cdot 283^{-1}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
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magma: Rank(E);
sage: E.rank()
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| Rank: | \(0\) | ||
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magma: Regulator(E);
sage: E.regulator()
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| Regulator: | \(1\) | ||
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magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(0.898993462968263\) | ||
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magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
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| Tamagawa product: | \( 2 \) = \( 2\cdot1\cdot1\cdot1 \) | ||
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magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(1\) | ||
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magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 498080.2.a.f
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)
\( q - q^{5} - 2q^{7} - 3q^{9} + q^{11} - 2q^{13} + 2q^{17} - 2q^{19} + O(q^{20}) \)
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magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 161280 | ||
| \( \Gamma_0(N) \)-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])
\( L(E,1) \) ≈ \( 1.7979869259365266 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \( I_3^{*} \) | Additive | 1 | 5 | 12 | 0 |
| \(5\) | \(1\) | \( I_{3} \) | Non-split multiplicative | 1 | 1 | 3 | 3 |
| \(11\) | \(1\) | \( I_{1} \) | Split multiplicative | -1 | 1 | 1 | 1 |
| \(283\) | \(1\) | \( I_{1} \) | 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]
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 no rational isogenies. Its isogeny class 498080.f consists of this curve only.