Base field \(\Q(\sqrt{-1}) \)
Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))
gp: K = nfinit(Polrev([1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([1,1]),K([0,1]),K([1,0]),K([1,1])])
gp: E = ellinit([Polrev([0,0]),Polrev([1,1]),Polrev([0,1]),Polrev([1,0]),Polrev([1,1])], K);
magma: E := EllipticCurve([K![0,0],K![1,1],K![0,1],K![1,0],K![1,1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-80i+277)\) | = | \((-4i+9)\cdot(4i+29)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 83129 \) | = | \(97\cdot857\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-80i+277)\) | = | \((-4i+9)\cdot(4i+29)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 83129 \) | = | \(97\cdot857\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{55140352}{83129} i - \frac{4861952}{83129} \) | ||
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: | \(3\) | ||
Generators | $\left(-i - \frac{3}{4} : \frac{3}{8} : 1\right)$ | $\left(-\frac{1}{2} i : -\frac{1}{4} i + \frac{3}{4} : 1\right)$ | $\left(i : 0 : 1\right)$ |
Heights | \(1.7689023969175415805866866939351903557\) | \(1.0819526677246979471786878324132700593\) | \(0.73696367541401498290413941990348618304\) |
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: | \( 3 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(3\) | ||
Regulator: | \( 0.078013512931511852939393770696018050162 \) | ||
Period: | \( 5.3488667588869796726449581921080309545 \) | ||
Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.3382710885069062482590681983424445821 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-4i+9)\) | \(97\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
\((4i+29)\) | \(857\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 83129.2-a consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.