Base field \(\Q(\sqrt{23}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 23 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-23, 0, 1]))
gp: K = nfinit(Polrev([-23, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,1]),K([-5019,-1047]),K([191446,39920])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([1,1]),Polrev([-5019,-1047]),Polrev([191446,39920])], K);
magma: E := EllipticCurve([K![1,0],K![1,0],K![1,1],K![-5019,-1047],K![191446,39920]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-7a+35)\) | = | \((-a+5)\cdot(-a+4)\cdot(-a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 98 \) | = | \(2\cdot7\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((3332a-20972)\) | = | \((-a+5)^{5}\cdot(-a+4)^{2}\cdot(-a-4)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 184473632 \) | = | \(2^{5}\cdot7^{2}\cdot7^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{19905325462039}{941192} a + \frac{95464498526147}{941192} \) | ||
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 | $\left(6 a + 29 : -4 a - 17 : 1\right)$ |
Height | \(0.85914466125252646333721391552108497964\) |
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: | $\left(6 a + \frac{115}{4} : -\frac{7}{2} a - \frac{119}{8} : 1\right)$ |
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: | \( 0.85914466125252646333721391552108497964 \) | ||
Period: | \( 12.398096436896503950798286673021105128 \) | ||
Tamagawa product: | \( 4 \) = \(1\cdot2\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.2210451538330734646708005563340842049 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a+5)\) | \(2\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
\((-a+4)\) | \(7\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((-a-4)\) | \(7\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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
98.1-h
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.