Base field \(\Q(\sqrt{-7}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
gp: K = nfinit(Polrev([2, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([1,1]),K([1,1]),K([-185,-22]),K([1039,-1443])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([1,1]),Polrev([-185,-22]),Polrev([1039,-1443])], K);
magma: E := EllipticCurve([K![1,1],K![1,1],K![1,1],K![-185,-22],K![1039,-1443]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((26a-174)\) | = | \((a)\cdot(-a+1)^{4}\cdot(-2a+1)\cdot(2a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 27104 \) | = | \(2\cdot2^{4}\cdot7\cdot11^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((618278080a+1372460992)\) | = | \((a)^{6}\cdot(-a+1)^{18}\cdot(-2a+1)^{6}\cdot(2a+1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 3496747289987252224 \) | = | \(2^{6}\cdot2^{18}\cdot7^{6}\cdot11^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{9938375}{21952} \) | ||
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 | $\left(-4 a + 13 : 32 a - 60 : 1\right)$ | |
Height | \(0.35516059594305829286218515852184157015\) | |
Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
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 generators: | $\left(\frac{5}{4} a - \frac{67}{4} : \frac{53}{8} a + \frac{73}{8} : 1\right)$ | $\left(-6 a + 3 : 4 a - 8 : 1\right)$ |
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: | \( 0.35516059594305829286218515852184157015 \) | ||
Period: | \( 0.39592229624813766434553534739165594500 \) | ||
Tamagawa product: | \( 576 \) = \(( 2 \cdot 3 )\cdot2^{2}\cdot( 2 \cdot 3 )\cdot2^{2}\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 7.6532906647796310445779593777773786732 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(2\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
\((-a+1)\) | \(2\) | \(4\) | \(I_{10}^{*}\) | Additive | \(-1\) | \(4\) | \(18\) | \(6\) |
\((-2a+1)\) | \(7\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
\((2a+1)\) | \(11\) | \(4\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
\(3\) | 3Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
27104.15-j
consists of curves linked by isogenies of
degrees dividing 36.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.