Base field 5.5.38569.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} + 4 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 4, 0, -5, 0, 1]))
gp: K = nfinit(Polrev([-1, 4, 0, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 0, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,1,5,0,-1]),K([0,-4,4,1,-1]),K([0,0,0,0,0]),K([70,-173,-313,36,72]),K([-496,1418,1494,-336,-339])])
gp: E = ellinit([Polrev([-3,1,5,0,-1]),Polrev([0,-4,4,1,-1]),Polrev([0,0,0,0,0]),Polrev([70,-173,-313,36,72]),Polrev([-496,1418,1494,-336,-339])], K);
magma: E := EllipticCurve([K![-3,1,5,0,-1],K![0,-4,4,1,-1],K![0,0,0,0,0],K![70,-173,-313,36,72],K![-496,1418,1494,-336,-339]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^3-3a-1)\) | = | \((a^3-3a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 17 \) | = | \(17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((22a^4+61a^3-130a^2-191a+150)\) | = | \((a^3-3a-1)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 6975757441 \) | = | \(17^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{63035509038380061495748094}{6975757441} a^{4} + \frac{128497588344189495041715327}{6975757441} a^{3} - \frac{53235816441571130890411893}{6975757441} a^{2} - \frac{108520961136492735262181368}{6975757441} a + \frac{30922567893134525558722397}{6975757441} \) | ||
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 | $\left(-\frac{124168}{32041} a^{4} - \frac{56131}{32041} a^{3} + \frac{496461}{32041} a^{2} + \frac{180140}{32041} a + \frac{65103}{32041} : \frac{21429659}{5735339} a^{4} + \frac{63107445}{5735339} a^{3} - \frac{86190463}{5735339} a^{2} - \frac{279886281}{5735339} a + \frac{52422108}{5735339} : 1\right)$ | |
Height | \(2.6128105177820355837973489989957110106\) | |
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: | $\left(6 a^{4} + 2 a^{3} - 26 a^{2} - 10 a + 4 : -2 a^{4} + 9 a^{2} + 2 a - 2 : 1\right)$ | $\left(-2 a^{4} - 2 a^{3} + 10 a^{2} + 10 a - 4 : -a^{2} - 2 a : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(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: | \( 2.6128105177820355837973489989957110106 \) | ||
Period: | \( 192.01825934281905933148811662306894564 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 1.59665569 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^3-3a-1)\) | \(17\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
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 and 4.
Its isogeny class
17.1-c
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.