Base field \(\Q(\zeta_{21})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Invariants
Conductor: | \((1)\) | = | \((1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 1 \) | = | 1 |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((1)\) | = | \((1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 1 \) | = | 1 |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( 7604567359488000 a^{5} + 7604567359488000 a^{4} - 38022836797440000 a^{3} - 30418269437952000 a^{2} + 45627404156928000 a - 6017401737216000 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z[(1+\sqrt{-147})/2]\) | (potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $N(\mathrm{U}(1))$ |
Mordell-Weil group
Rank: | \(0\) |
Torsion structure: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 0.18837610010410643558659577912015378451 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 0.671415 \) | ||
Analytic order of Ш: | \( 2401 \) (rounded) |
Local data at primes of bad reduction
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(7\) | 7B.1.6[3] |
For all other primes \(p\), the image is a Borel subgroup if \(p=3\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 7, 21, 49 and 147.
Its isogeny class
1.1-a
consists of curves linked by isogenies of
degrees dividing 147.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:
Base field | Curve |
---|---|
\(\Q(\sqrt{21}) \) | a curve with conductor norm 49 (not in the database) |