Base field \(\Q(\sqrt{33}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 8 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-8, -1, 1]))
gp: K = nfinit(Polrev([-8, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([0,1]),K([0,0]),K([19445,-5765]),K([9955158,-2952054])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,1]),Polrev([0,0]),Polrev([19445,-5765]),Polrev([9955158,-2952054])], K);
magma: E := EllipticCurve([K![0,0],K![0,1],K![0,0],K![19445,-5765],K![9955158,-2952054]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-16a+56)\) | = | \((-a-2)^{3}\cdot(-a+3)^{3}\cdot(-2a+7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 192 \) | = | \(2^{3}\cdot2^{3}\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-13436928)\) | = | \((-a-2)^{11}\cdot(-a+3)^{11}\cdot(-2a+7)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 180551034077184 \) | = | \(2^{11}\cdot2^{11}\cdot3^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{207646}{6561} \) | ||
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: | \(0\) |
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);
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Torsion generator: | $\left(37 a - 126 : 0 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 2.8417542587507934505185738139717168448 \) | ||
Tamagawa product: | \( 2 \) = \(1\cdot1\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 0.98937184033091950994975185688833795130 \) | ||
Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a-2)\) | \(2\) | \(1\) | \(II^{*}\) | Additive | \(1\) | \(3\) | \(11\) | \(0\) |
\((-a+3)\) | \(2\) | \(1\) | \(II^{*}\) | Additive | \(1\) | \(3\) | \(11\) | \(0\) |
\((-2a+7)\) | \(3\) | \(2\) | \(I_{16}\) | 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 and 8.
Its isogeny class
192.1-c
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 72.a6 |
\(\Q\) | 2904.c6 |