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([0,0]),K([0,0]),K([-133,240]),K([-1680,-246])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-133,240]),Polrev([-1680,-246])], K);
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![-133,240],K![-1680,-246]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((80)\) | = | \((i+1)^{8}\cdot(-i-2)\cdot(2i+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 6400 \) | = | \(2^{8}\cdot5\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-287447040i-2513438720)\) | = | \((i+1)^{23}\cdot(-i-2)\cdot(2i+1)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 6400000000000000000 \) | = | \(2^{23}\cdot5\cdot5^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{22845545233191}{152587890625} i + \frac{135893651813613}{152587890625} \) | ||
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 | $\left(\frac{7522}{289} i - \frac{593}{289} : -\frac{552331}{4913} i + \frac{469867}{4913} : 1\right)$ |
Height | \(2.8888213522577024586640307688620386757\) |
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: | $\left(-6 i + 16 : 0 : 1\right)$ |
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: | \( 2.8888213522577024586640307688620386757 \) | ||
Period: | \( 0.37461112273012781001754472109836674726 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot1\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.1643692202720479172762592081405667042 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((i+1)\) | \(2\) | \(2\) | \(I_{11}^{*}\) | Additive | \(1\) | \(8\) | \(23\) | \(0\) |
\((-i-2)\) | \(5\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((2i+1)\) | \(5\) | \(2\) | \(I_{16}\) | Non-split multiplicative | \(1\) | \(1\) | \(16\) | \(16\) |
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, 4, 8 and 16.
Its isogeny class
6400.2-a
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.