Base field \(\Q(\sqrt{-11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
gp: K = nfinit(Polrev([3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([1,-1]),K([0,0]),K([-355,-226]),K([-5243,-2143])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,-1]),Polrev([0,0]),Polrev([-355,-226]),Polrev([-5243,-2143])], K);
magma: E := EllipticCurve([K![1,0],K![1,-1],K![0,0],K![-355,-226],K![-5243,-2143]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-99a+33)\) | = | \((-a)\cdot(a-1)\cdot(a-2)^{2}\cdot(-2a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 27225 \) | = | \(3\cdot3\cdot5^{2}\cdot11^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-4171618869a-13460658552)\) | = | \((-a)^{20}\cdot(a-1)\cdot(a-2)^{6}\cdot(-2a+1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 289549277822810671875 \) | = | \(3^{20}\cdot3\cdot5^{6}\cdot11^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{2927543402641}{3486784401} a + \frac{1635099303025}{3486784401} \) | ||
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 | $\left(\frac{25}{9} a - \frac{103}{3} : -\frac{2672}{27} a + \frac{50}{9} : 1\right)$ |
Height | \(2.8442268042642406084306206819437615364\) |
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: | $\left(-7 a - \frac{53}{4} : \frac{7}{2} a + \frac{53}{8} : 1\right)$ |
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: | \( 2.8442268042642406084306206819437615364 \) | ||
Period: | \( 0.26737169462912409322884330718055061322 \) | ||
Tamagawa product: | \( 16 \) = \(2\cdot1\cdot2^{2}\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 3.6686247670926623025179502049052719676 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(3\) | \(2\) | \(I_{20}\) | Non-split multiplicative | \(1\) | \(1\) | \(20\) | \(20\) |
\((a-1)\) | \(3\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
\((a-2)\) | \(5\) | \(4\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
\((-2a+1)\) | \(11\) | \(2\) | \(I_0^{*}\) | 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 |
---|---|
\(2\) | 2B |
\(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 5, 10 and 20.
Its isogeny class
27225.6-c
consists of curves linked by isogenies of
degrees dividing 20.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.