Base field \(\Q(\sqrt{11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 11 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-11, 0, 1]))
gp: K = nfinit(Polrev([-11, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([1,-1]),K([0,0]),K([-665,200]),K([-9409,2837])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,-1]),Polrev([0,0]),Polrev([-665,200]),Polrev([-9409,2837])], K);
magma: E := EllipticCurve([K![1,1],K![1,-1],K![0,0],K![-665,200],K![-9409,2837]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((10a+30)\) | = | \((a+3)^{3}\cdot(a-4)\cdot(-a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 200 \) | = | \(2^{3}\cdot5\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-25000a+162500)\) | = | \((a+3)^{4}\cdot(a-4)^{8}\cdot(-a-4)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 19531250000 \) | = | \(2^{4}\cdot5^{8}\cdot5^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{556737823072}{390625} a + \frac{1849158305968}{390625} \) | ||
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 | $\left(-8 a + 29 : 37 a - 123 : 1\right)$ |
Height | \(0.27031373928760319808943819581043902296\) |
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: | $\left(\frac{5}{2} a - 8 : \frac{11}{4} a - \frac{39}{4} : 1\right)$ |
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: | \( 0.27031373928760319808943819581043902296 \) | ||
Period: | \( 7.1099223357107386019584765377417871487 \) | ||
Tamagawa product: | \( 20 \) = \(2\cdot2\cdot5\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.8973878778800196041165186211795156522 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a+3)\) | \(2\) | \(2\) | \(III\) | Additive | \(1\) | \(3\) | \(4\) | \(0\) |
\((a-4)\) | \(5\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
\((-a-4)\) | \(5\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
200.1-g
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.