Base field \(\Q(\zeta_{15})^+\)
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 4 x^{2} + 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 4, -4, -1, 1]))
gp: K = nfinit(Polrev([1, 4, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,0,1,0]),K([-1,-4,1,1]),K([1,1,0,0]),K([-4,4,2,-2]),K([6,-6,-1,1])])
gp: E = ellinit([Polrev([-1,0,1,0]),Polrev([-1,-4,1,1]),Polrev([1,1,0,0]),Polrev([-4,4,2,-2]),Polrev([6,-6,-1,1])], K);
magma: E := EllipticCurve([K![-1,0,1,0],K![-1,-4,1,1],K![1,1,0,0],K![-4,4,2,-2],K![6,-6,-1,1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((4a^3+a^2-12a+1)\) | = | \((-a-1)\cdot(a^2-a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 145 \) | = | \(5\cdot29\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((45a^3+10a^2-140a-5)\) | = | \((-a-1)^{7}\cdot(a^2-a-3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 2265625 \) | = | \(5^{7}\cdot29\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{46928367}{725} a^{3} - \frac{15506749}{725} a^{2} + \frac{166828287}{725} a + \frac{34453039}{725} \) | ||
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: | \(\Z/14\Z\) |
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: | $\left(-a^{2} + 2 : a^{3} - 4 a + 2 : 1\right)$ |
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: | \( 1003.1162302038611365004072727086735381 \) | ||
Tamagawa product: | \( 7 \) = \(7\cdot1\) | ||
Torsion order: | \(14\) | ||
Leading coefficient: | \( 1.06811241908055 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a-1)\) | \(5\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
\((a^2-a-3)\) | \(29\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 |
\(7\) | 7B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 7 and 14.
Its isogeny class
145.2-d
consists of curves linked by isogenies of
degrees dividing 14.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.