Base field \(\Q(\sqrt{29}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 7 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-7, -1, 1]))
gp: K = nfinit(Polrev([-7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([1,0]),K([-92951,29048]),K([-14040061,4397257])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([1,0]),Polrev([-92951,29048]),Polrev([-14040061,4397257])], K);
magma: E := EllipticCurve([K![0,1],K![1,0],K![1,0],K![-92951,29048],K![-14040061,4397257]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((30)\) | = | \((2)\cdot(-a-1)\cdot(-a+2)\cdot(3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 900 \) | = | \(4\cdot5\cdot5\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((309586821120)\) | = | \((2)^{20}\cdot(-a-1)\cdot(-a+2)\cdot(3)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 95843999811186878054400 \) | = | \(4^{20}\cdot5\cdot5\cdot9^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{21685195471991381}{309586821120} \) | ||
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: | \(2\) | |
Generators | $\left(-\frac{83883043}{648025} a + \frac{37304863}{92575} : -\frac{1419392877351}{521660125} a + \frac{648708477066}{74522875} : 1\right)$ | $\left(-\frac{19081}{196} a + \frac{4211}{14} : \frac{1310061}{1372} a - \frac{1187577}{392} : 1\right)$ |
Heights | \(4.8115149076508180749335169761068050056\) | \(4.8115149076508180749335169761068050056\) |
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);
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Torsion generator: | $\left(37 a - 127 : 45 a - 130 : 1\right)$ | |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 23.150675706546060387998884095611079135 \) | ||
Period: | \( 0.29955507000793312798641123075071442828 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot1\cdot1\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 5.1511160979271774260856007935407696917 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2)\) | \(4\) | \(2\) | \(I_{20}\) | Non-split multiplicative | \(1\) | \(1\) | \(20\) | \(20\) |
\((-a-1)\) | \(5\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
\((-a+2)\) | \(5\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
\((3)\) | \(9\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
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 |
\(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
900.1-f
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.