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([-1,0]),K([0,1]),K([21,4]),K([31,20])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([0,1]),Polrev([21,4]),Polrev([31,20])], K);
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,1],K![21,4],K![31,20]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((162)\) | = | \((a)\cdot(-a+1)\cdot(3)^{4}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 26244 \) | = | \(2\cdot2\cdot9^{4}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-1306368a-559872)\) | = | \((a)^{8}\cdot(-a+1)^{15}\cdot(3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 4458050224128 \) | = | \(2^{8}\cdot2^{15}\cdot9^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{3499281}{32768} a + \frac{12237507}{16384} \) | ||
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 | $\left(-3 a + 2 : a - 9 : 1\right)$ |
Height | \(0.27360422669038536884710404715132468558\) |
Torsion structure: | \(\Z/3\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(1 : -2 a - 8 : 1\right)$ |
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: | \( 0.27360422669038536884710404715132468558 \) | ||
Period: | \( 1.2231138324690880353449851609753281590 \) | ||
Tamagawa product: | \( 90 \) = \(2\cdot( 3 \cdot 5 )\cdot3\) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 5.0594190449799024760242457883855773894 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(2\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
\((-a+1)\) | \(2\) | \(15\) | \(I_{15}\) | Split multiplicative | \(-1\) | \(1\) | \(15\) | \(15\) |
\((3)\) | \(9\) | \(3\) | \(IV\) | Additive | \(-1\) | \(4\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
26244.2-d
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.