Base field \(\Q(\sqrt{-11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
gp: K = nfinit(Polrev([3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([1,1]),K([1,0]),K([-21896,3646]),K([-1131650,302918])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([1,0]),Polrev([-21896,3646]),Polrev([-1131650,302918])], K);
magma: E := EllipticCurve([K![1,1],K![1,1],K![1,0],K![-21896,3646],K![-1131650,302918]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-55a-110)\) | = | \((a-1)^{2}\cdot(-a-1)\cdot(a-2)\cdot(-2a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 27225 \) | = | \(3^{2}\cdot5\cdot5\cdot11^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-12402102306275a+96762336789800)\) | = | \((a-1)^{3}\cdot(-a-1)^{2}\cdot(a-2)^{24}\cdot(-2a+1)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 8624329845607280731201171875 \) | = | \(3^{3}\cdot5^{2}\cdot5^{24}\cdot11^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{2638659751550362590877}{655651092529296875} a + \frac{638610383092162643063}{59604644775390625} \) | ||
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: | $\left(-\frac{63}{4} a + \frac{343}{2} : -70 a - \frac{879}{8} : 1\right)$ |
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: | \( 0.058442987389506661254191517095982846832 \) | ||
Tamagawa product: | \( 384 \) = \(2\cdot2\cdot( 2^{3} \cdot 3 )\cdot2^{2}\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 3.3832749521186759124801279025285661095 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a-1)\) | \(3\) | \(2\) | \(III\) | Additive | \(1\) | \(2\) | \(3\) | \(0\) |
\((-a-1)\) | \(5\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((a-2)\) | \(5\) | \(24\) | \(I_{24}\) | Split multiplicative | \(-1\) | \(1\) | \(24\) | \(24\) |
\((-2a+1)\) | \(11\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
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
27225.8-f
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.