Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
gp: K = nfinit(Polrev([1, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([0,0]),K([-4,0])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-4,0])], K);
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![0,0],K![-4,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((144)\) | = | \((-2a+1)^{4}\cdot(2)^{4}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 20736 \) | = | \(3^{4}\cdot4^{4}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-6912)\) | = | \((-2a+1)^{6}\cdot(2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 47775744 \) | = | \(3^{6}\cdot4^{8}\) |
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: | \(\Z[(1+\sqrt{-3})/2]\) | (complex multiplication) | |
Geometric endomorphism ring: | \(\Z[(1+\sqrt{-3})/2]\) | ||
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{U}(1)$ |
Mordell-Weil group
Rank: | \(2\) | |
Generators | $\left(-2 a : -2 : 1\right)$ | $\left(-2 : -4 a + 2 : 1\right)$ |
Heights | \(0.45032068563987499353186356396948461475\) | \(0.15010689521329166451062118798982820492\) |
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: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 0.016899059992930593017726562187830141623 \) | ||
Period: | \( 3.2179112590980491572484561477941328708 \) | ||
Tamagawa product: | \( 12 \) = \(3\cdot2^{2}\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.0140275432175382937911613215189897843 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-2a+1)\) | \(3\) | \(3\) | \(IV\) | Additive | \(-1\) | \(4\) | \(6\) | \(0\) |
\((2)\) | \(4\) | \(4\) | \(I_0^{*}\) | Additive | \(1\) | \(4\) | \(8\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
The image is a Borel subgroup if \(p=3\), a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has no rational isogenies other than endomorphisms. Its isogeny class 20736.1-CMd consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 432.d2 |
\(\Q\) | 432.d1 |