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([1,1]),K([1,1]),K([-28998,12675]),K([-1781967,1245458])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([1,1]),Polrev([-28998,12675]),Polrev([-1781967,1245458])], K);
magma: E := EllipticCurve([K![1,1],K![1,1],K![1,1],K![-28998,12675],K![-1781967,1245458]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((100i-240)\) | = | \((i+1)^{4}\cdot(-i-2)\cdot(2i+1)\cdot(2i+3)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
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: | \((-5938128000i+8070704000)\) | = | \((i+1)^{15}\cdot(-i-2)^{8}\cdot(2i+1)^{3}\cdot(2i+3)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 100397627200000000000 \) | = | \(2^{15}\cdot5^{8}\cdot5^{3}\cdot13^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{4023422266102893}{20312500} i + \frac{5856979210600901}{20312500} \) | ||
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 | $\left(\frac{57406}{2809} i - \frac{283071}{2809} : \frac{6042385}{148877} i + \frac{9315284}{148877} : 1\right)$ |
Height | \(4.2376886255894756760173462412392675864\) |
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: | $\left(\frac{41}{2} i - 101 : \frac{159}{4} i + \frac{241}{4} : 1\right)$ |
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: | \( 4.2376886255894756760173462412392675864 \) | ||
Period: | \( 0.10029150424693393683257575123452340354 \) | ||
Tamagawa product: | \( 16 \) = \(2^{2}\cdot2\cdot1\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 3.4000333343239243000530617954080781598 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((i+1)\) | \(2\) | \(4\) | \(I_{7}^{*}\) | Additive | \(1\) | \(4\) | \(15\) | \(3\) |
\((-i-2)\) | \(5\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
\((2i+1)\) | \(5\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
\((2i+3)\) | \(13\) | \(2\) | \(I_{1}^{*}\) | Additive | \(1\) | \(2\) | \(7\) | \(1\) |
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 |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
67600.6-a
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.