Base field 5.5.24217.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} - x^{2} + 3 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))
gp: K = nfinit(Polrev([1, 3, -1, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,-3,5,1,-1]),K([0,-4,-1,1,0]),K([0,1,-4,0,1]),K([-45,-110,-51,22,13]),K([-125,-393,-298,75,72])])
gp: E = ellinit([Polrev([-2,-3,5,1,-1]),Polrev([0,-4,-1,1,0]),Polrev([0,1,-4,0,1]),Polrev([-45,-110,-51,22,13]),Polrev([-125,-393,-298,75,72])], K);
magma: E := EllipticCurve([K![-2,-3,5,1,-1],K![0,-4,-1,1,0],K![0,1,-4,0,1],K![-45,-110,-51,22,13],K![-125,-393,-298,75,72]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^4-a^3-5a^2+5a+4)\) | = | \((a^4-a^3-5a^2+5a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 97 \) | = | \(97\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-3a^4+13a^2+4a-9)\) | = | \((a^4-a^3-5a^2+5a+4)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 9409 \) | = | \(97^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{2699246979785}{9409} a^{4} - \frac{2388620148409}{9409} a^{3} + \frac{11413457143722}{9409} a^{2} + \frac{12792778212781}{9409} a + \frac{3119261293489}{9409} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(0\) | |
Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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Torsion generators: | $\left(4 a^{4} - a^{3} - 18 a^{2} + 6 : -a^{4} + a^{3} + 4 a^{2} - 4 a - 1 : 1\right)$ | $\left(\frac{1}{2} a^{4} + \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{11}{4} a - \frac{13}{4} : -\frac{3}{2} a^{4} + \frac{9}{8} a^{3} + \frac{53}{8} a^{2} - \frac{9}{2} a - \frac{19}{8} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 299.55357568807473204446513019388583313 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 0.962463662 \) | ||
Analytic order of Ш: | \( 4 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^4-a^3-5a^2+5a+4)\) | \(97\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
97.1-a
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.