Base field \(\Q(\sqrt{217}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 54 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-54, -1, 1]))
gp: K = nfinit(Polrev([-54, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-54, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,1]),K([-2073057,-301955]),K([12103884350,1763011434])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,1]),Polrev([-2073057,-301955]),Polrev([12103884350,1763011434])], K);
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,1],K![-2073057,-301955],K![12103884350,1763011434]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-996a+7834)\) | = | \((-a+8)\cdot(-a-7)\cdot(-498a+3917)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 28 \) | = | \(2\cdot2\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((1344a-11200)\) | = | \((-a+8)^{6}\cdot(-a-7)^{12}\cdot(-498a+3917)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 12845056 \) | = | \(2^{6}\cdot2^{12}\cdot7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{3360549}{28672} a + \frac{12727843}{14336} \) | ||
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 | $\left(-179 a - 1229 : -9358 a - 64244 : 1\right)$ |
Height | \(0.14906175771795617237507129230107754112\) |
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: | $\left(-\frac{939}{4} a - \frac{6447}{4} : \frac{935}{8} a + \frac{6443}{8} : 1\right)$ |
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: | \( 0.14906175771795617237507129230107754112 \) | ||
Period: | \( 9.6524497633838462667163221756553188592 \) | ||
Tamagawa product: | \( 48 \) = \(2\cdot( 2^{2} \cdot 3 )\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.3441487289522040673643404290047398609 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a+8)\) | \(2\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
\((-a-7)\) | \(2\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
\((-498a+3917)\) | \(7\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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 |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
28.1-d
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.