Base field 5.5.65657.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([1, 5, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,-2,5,1,-1]),K([-3,-4,5,1,-1]),K([0,1,0,0,0]),K([-6,-20,21,5,-6]),K([-59,-325,-275,191,148])])
gp: E = ellinit([Polrev([-3,-2,5,1,-1]),Polrev([-3,-4,5,1,-1]),Polrev([0,1,0,0,0]),Polrev([-6,-20,21,5,-6]),Polrev([-59,-325,-275,191,148])], K);
magma: E := EllipticCurve([K![-3,-2,5,1,-1],K![-3,-4,5,1,-1],K![0,1,0,0,0],K![-6,-20,21,5,-6],K![-59,-325,-275,191,148]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^4+a^3+3a^2-2a)\) | = | \((-a^2+a+2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 25 \) | = | \(5^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-561a^4+517a^3+2290a^2-525a-1630)\) | = | \((-a^2+a+2)^{21}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -476837158203125 \) | = | \(-5^{21}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{175802229236840746}{30517578125} a^{4} + \frac{65933855205252401}{30517578125} a^{3} - \frac{788398234528339998}{30517578125} a^{2} - \frac{732499454042744644}{30517578125} a - \frac{127944679222457853}{30517578125} \) | ||
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: | \(1\) |
Generator | $\left(4 a^{3} - a^{2} - 10 a + 5 : 13 a^{4} - 13 a^{3} - 37 a^{2} + 41 a + 1 : 1\right)$ |
Height | \(0.13409633773563903292072676189736247852\) |
Torsion structure: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.13409633773563903292072676189736247852 \) | ||
Period: | \( 271.93917715149453964539659291145143449 \) | ||
Tamagawa product: | \( 4 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.84628363 \) | ||
Analytic order of Ш: | \( 1 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^2+a+2)\) | \(5\) | \(4\) | \(I_{15}^{*}\) | Additive | \(1\) | \(2\) | \(21\) | \(15\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3B |
\(5\) | 5B.4.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
25.1-c
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.