Base field \(\Q(\sqrt{26}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 26 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-26, 0, 1]))
gp: K = nfinit(Polrev([-26, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-26, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([-1,1]),K([0,1]),K([-7517,1479]),K([311088,-61002])])
gp: E = ellinit([Polrev([0,1]),Polrev([-1,1]),Polrev([0,1]),Polrev([-7517,1479]),Polrev([311088,-61002])], K);
magma: E := EllipticCurve([K![0,1],K![-1,1],K![0,1],K![-7517,1479],K![311088,-61002]]);
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a-4)\) | = | \((2,a)\cdot(5,a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 10 \) | = | \(2\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-4096a-16384)\) | = | \((2,a)^{25}\cdot(5,a+4)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -167772160 \) | = | \(-2^{25}\cdot5\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((-64a-256)\) | = | \((2,a)^{13}\cdot(5,a+4)\) |
Minimal discriminant norm: | \( -40960 \) | = | \(-2^{13}\cdot5\) |
j-invariant: | \( -\frac{391019}{640} a + \frac{357337}{80} \) | ||
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 | $\left(5 a - 29 : -97 a + 501 : 1\right)$ |
Height | \(0.53669898327762832678930604337895674454\) |
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: | \( 0.53669898327762832678930604337895674454 \) | ||
Period: | \( 17.175006006286280613179036547465914024 \) | ||
Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.8077609306629443636044875844950855053 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(1\) | \(I_{13}\) | Non-split multiplicative | \(1\) | \(1\) | \(13\) | \(13\) |
\((5,a+4)\) | \(5\) | \(1\) | \(I_{1}\) | 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 10.1-b 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.