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([1,0]),K([0,1]),K([0,0]),K([190,424]),K([-7320,1060])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,1]),Polrev([0,0]),Polrev([190,424]),Polrev([-7320,1060])], K);
magma: E := EllipticCurve([K![1,0],K![0,1],K![0,0],K![190,424],K![-7320,1060]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((66)\) | = | \((a)\cdot(-a+1)\cdot(3)\cdot(-2a+3)\cdot(2a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 4356 \) | = | \(2\cdot2\cdot9\cdot11\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((47509836a+1497479676)\) | = | \((a)^{2}\cdot(-a+1)^{42}\cdot(3)\cdot(-2a+3)^{3}\cdot(2a+1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 2318104762886651904 \) | = | \(2^{2}\cdot2^{42}\cdot9\cdot11^{3}\cdot11\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{105273133194758264123}{17561399718838272} a + \frac{223715565502137382049}{8780699859419136} \) | ||
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: | \(0\) |
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: | $\left(20 a - 24 : -10 a + 12 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 0.34868622566156944733249116100389497845 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\cdot1\cdot1\cdot1\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.3722380994995284318041997749371534826 \) | ||
Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(2\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((-a+1)\) | \(2\) | \(2\) | \(I_{42}\) | Non-split multiplicative | \(1\) | \(1\) | \(42\) | \(42\) |
\((3)\) | \(9\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
\((-2a+3)\) | \(11\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
\((2a+1)\) | \(11\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
4356.5-c
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.