Base field \(\Q(\sqrt{409}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 102 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-102, -1, 1]))
gp: K = nfinit(Polrev([-102, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-102, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([-1,0]),K([0,0]),K([-517506398338217684932,48766729438261090337]),K([-6253390946192205276990784538424,589282423026818412281150467642])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,0]),Polrev([0,0]),Polrev([-517506398338217684932,48766729438261090337]),Polrev([-6253390946192205276990784538424,589282423026818412281150467642])], K);
magma: E := EllipticCurve([K![1,1],K![-1,0],K![0,0],K![-517506398338217684932,48766729438261090337],K![-6253390946192205276990784538424,589282423026818412281150467642]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-438a-4210)\) | = | \((-219a+2324)\cdot(-219a-2105)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 8 \) | = | \(2\cdot2^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((3504a-37184)\) | = | \((-219a+2324)^{5}\cdot(-219a-2105)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 512 \) | = | \(2^{5}\cdot2^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{1622025}{32} a + \frac{69656571}{32} \) | ||
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 | $\left(\frac{203488023512}{225} a - \frac{2159389310813}{225} : -\frac{8317962653802823}{3375} a + \frac{88269173349862702}{3375} : 1\right)$ |
Height | \(2.4357594982320768824284675543888288903\) |
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);
| |
Torsion generator: | $\left(\frac{3638773761}{4} a - \frac{38614209467}{4} : \frac{31336661945}{8} a - \frac{332540714155}{8} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 2.4357594982320768824284675543888288903 \) | ||
Period: | \( 18.913185199581039667292312335183660353 \) | ||
Tamagawa product: | \( 3 \) = \(1\cdot3\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 3.4168718041580988402382419603504051738 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-219a+2324)\) | \(2\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
\((-219a-2105)\) | \(2\) | \(3\) | \(IV\) | Additive | \(-1\) | \(2\) | \(4\) | \(0\) |
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.
Its isogeny class
8.2-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.