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([0,0]),K([1,-1]),K([1,0]),K([1118307,-602166]),K([506323495,81612670])])
gp: E = ellinit([Polrev([0,0]),Polrev([1,-1]),Polrev([1,0]),Polrev([1118307,-602166]),Polrev([506323495,81612670])], K);
magma: E := EllipticCurve([K![0,0],K![1,-1],K![1,0],K![1118307,-602166],K![506323495,81612670]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-77a+143)\) | = | \((a-1)^{2}\cdot(-a-1)^{2}\cdot(-2a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 27225 \) | = | \(3^{2}\cdot5^{2}\cdot11^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-13938232a-36704987)\) | = | \((a-1)^{6}\cdot(-a-1)^{6}\cdot(-2a+1)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 2441681628890625 \) | = | \(3^{6}\cdot5^{6}\cdot11^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{52893159101157376}{11} \) | ||
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 | $\left(-\frac{430942713883413617448205997457330940988}{1596071523437224762559114913469364025} a - \frac{6819096374545310445433686495595513222}{40924910857364737501515767012034975} : -\frac{4819259164280776436339161660444331038259973850591782855557}{2016408519478354698778108912838232431460063559394569875} a - \frac{2461716432364949821940830084937449944323547157695726671104}{672136173159451566259369637612744143820021186464856625} : 1\right)$ |
Height | \(47.862984316499901555613487424180986975\) |
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: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 47.862984316499901555613487424180986975 \) | ||
Period: | \( 0.028828495118589295723683022995385794839 \) | ||
Tamagawa product: | \( 4 \) = \(1\cdot1\cdot2^{2}\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 6.6564915693413781646984953788195755538 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a-1)\) | \(3\) | \(1\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
\((-a-1)\) | \(5\) | \(1\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
\((-2a+1)\) | \(11\) | \(4\) | \(I_{2}^{*}\) | 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 |
---|---|
\(2\) | 2Cn |
\(5\) | 5B.4.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5 and 25.
Its isogeny class
27225.7-e
consists of curves linked by isogenies of
degrees dividing 25.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.