Base field \(\Q(\zeta_{16})^+\)
Generator \(a\), with minimal polynomial \( x^{4} - 4 x^{2} + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -4, 0, 1]))
gp: K = nfinit(Polrev([2, 0, -4, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -4, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,0,0]),K([1,3,0,-1]),K([1,0,0,0]),K([-8,-12,2,3]),K([7,9,-3,-3])])
gp: E = ellinit([Polrev([0,1,0,0]),Polrev([1,3,0,-1]),Polrev([1,0,0,0]),Polrev([-8,-12,2,3]),Polrev([7,9,-3,-3])], K);
magma: E := EllipticCurve([K![0,1,0,0],K![1,3,0,-1],K![1,0,0,0],K![-8,-12,2,3],K![7,9,-3,-3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^3-a^2+3a+1)\) | = | \((-a^3-a^2+3a+1)\) |
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: | \((-125a^3-212a^2+317a+197)\) | = | \((-a^3-a^2+3a+1)^{8}\) |
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: | \( -\frac{113010507685176}{6975757441} a^{3} - \frac{82992167336432}{6975757441} a^{2} + \frac{372591802841328}{6975757441} a + \frac{292134350112776}{6975757441} \) | ||
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: | \(0\) |
Torsion structure: | \(\Z/2\Z\) |
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 generator: | $\left(-\frac{1}{4} a^{2} + a + 1 : \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2} : 1\right)$ |
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: | \( 152.71624493665289097231743275804190295 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.68729206862583 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^3-a^2+3a+1)\) | \(17\) | \(2\) | \(I_{8}\) | 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 |
---|---|
\(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
17.2-a
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.