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([1,1]),K([0,1]),K([0,0]),K([-17,39]),K([-215,30])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,1]),Polrev([0,0]),Polrev([-17,39]),Polrev([-215,30])], K);
magma: E := EllipticCurve([K![1,1],K![0,1],K![0,0],K![-17,39],K![-215,30]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((100i+240)\) | = | \((i+1)^{4}\cdot(-i-2)\cdot(2i+1)\cdot(-3i-2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 67600 \) | = | \(2^{4}\cdot5\cdot5\cdot13^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((8280000i-20350000)\) | = | \((i+1)^{8}\cdot(-i-2)^{4}\cdot(2i+1)^{4}\cdot(-3i-2)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 482680900000000 \) | = | \(2^{8}\cdot5^{4}\cdot5^{4}\cdot13^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{237276}{625} \) | ||
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 | $\left(-i + 3 : 14 i + 3 : 1\right)$ | |
Height | \(0.56462295606125061249017430824058113195\) | |
Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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Torsion generators: | $\left(5 i - 5 : 5 : 1\right)$ | $\left(-\frac{7}{2} i - 2 : \frac{11}{4} i - \frac{3}{4} : 1\right)$ |
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: | \( 0.56462295606125061249017430824058113195 \) | ||
Period: | \( 0.83118745314622114237037906877282822768 \) | ||
Tamagawa product: | \( 64 \) = \(2\cdot2\cdot2^{2}\cdot2^{2}\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 3.7544601346915329778737150557821264372 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((i+1)\) | \(2\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(4\) | \(8\) | \(0\) |
\((-i-2)\) | \(5\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
\((2i+1)\) | \(5\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
\((-3i-2)\) | \(13\) | \(4\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
67600.4-d
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.