Base field 4.4.4525.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 3 x + 9 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, 3, -7, -1, 1]))
gp: K = nfinit(Polrev([9, 3, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 3, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-1/3,-1/3,1/3]),K([3,10/3,-2/3,-1/3]),K([-3,-4/3,2/3,1/3]),K([-2,-1,3,2]),K([-17,-35/3,28/3,17/3])])
gp: E = ellinit([Polrev([0,-1/3,-1/3,1/3]),Polrev([3,10/3,-2/3,-1/3]),Polrev([-3,-4/3,2/3,1/3]),Polrev([-2,-1,3,2]),Polrev([-17,-35/3,28/3,17/3])], K);
magma: E := EllipticCurve([K![0,-1/3,-1/3,1/3],K![3,10/3,-2/3,-1/3],K![-3,-4/3,2/3,1/3],K![-2,-1,3,2],K![-17,-35/3,28/3,17/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-1/3a^3+4/3a^2+1/3a-5)\) | = | \((-1/3a^3-2/3a^2+4/3a+1)^{2}\) |
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: | \((-46/3a^3-17/3a^2+235/3a+27)\) | = | \((-1/3a^3-2/3a^2+4/3a+1)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -9765625 \) | = | \(-5^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{18442}{3} a^{3} - \frac{15974}{3} a^{2} + \frac{129295}{3} a + 47469 \) | ||
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: | 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: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 108.75237645179091213369254298347953959 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.61670009614169 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-1/3a^3-2/3a^2+4/3a+1)\) | \(5\) | \(1\) | \(II^{*}\) | Additive | \(1\) | \(2\) | \(10\) | \(0\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
25.3-a
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.