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([0,0]),K([81,108])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([81,108])], K);
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![0,0],K![81,108]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((216i+162)\) | = | \((i+1)^{2}\cdot(-i-2)^{2}\cdot(3)^{3}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 72900 \) | = | \(2^{2}\cdot5^{2}\cdot9^{3}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-7558272i+2204496)\) | = | \((i+1)^{8}\cdot(-i-2)^{4}\cdot(3)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 61987278240000 \) | = | \(2^{8}\cdot5^{4}\cdot9^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( 0 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z[(1+\sqrt{-3})/2]\) | (potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $N(\mathrm{U}(1))$ |
Mordell-Weil group
Rank: | \(1\) |
Generator | $\left(2 i + 3 : 7 i + 11 : 1\right)$ |
Height | \(2.1882000663041535905650168482666709866\) |
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: | \( 2.1882000663041535905650168482666709866 \) | ||
Period: | \( 0.99574806451908951450623412338301448163 \) | ||
Tamagawa product: | \( 1 \) = \(1\cdot1\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 4.3577919616058085660481931972976126601 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((i+1)\) | \(2\) | \(1\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
\((-i-2)\) | \(5\) | \(1\) | \(IV\) | Additive | \(-1\) | \(2\) | \(4\) | \(0\) |
\((3)\) | \(9\) | \(1\) | \(IV^{*}\) | Additive | \(1\) | \(3\) | \(9\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3B.1.2 |
For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
72900.1-c
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.