Base field 3.3.1825.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 8 x + 7 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([7, -8, -1, 1]))
gp: K = nfinit(Polrev([7, -8, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, -8, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-5,0,1]),K([-6,0,1]),K([-5,1,1]),K([-7,-1,1]),K([-2,-1,0])])
gp: E = ellinit([Polrev([-5,0,1]),Polrev([-6,0,1]),Polrev([-5,1,1]),Polrev([-7,-1,1]),Polrev([-2,-1,0])], K);
magma: E := EllipticCurve([K![-5,0,1],K![-6,0,1],K![-5,1,1],K![-7,-1,1],K![-2,-1,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^2-5)\) | = | \((a^2-5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 41 \) | = | \(41\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((a^2-5)\) | = | \((a^2-5)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -41 \) | = | \(-41\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{12575}{41} a^{2} - \frac{3030}{41} a - \frac{26442}{41} \) | ||
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 | $\left(-a + 1 : -a^{2} + 3 a : 1\right)$ |
Height | \(0.18879006602518194875041769412666242271\) |
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: | \( 0.18879006602518194875041769412666242271 \) | ||
Period: | \( 323.44954015900863157327823583148868928 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 4.2882045838919221197223113479911075353 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^2-5)\) | \(41\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 41.2-e consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.