Base field 3.3.1957.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 9 x + 10 \); class number \(2\).
Weierstrass equation
This is a global minimal model.
Invariants
Conductor: | \((11,a-4)\) | = | \((11,a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 11 \) | = | \(11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((4a^2-53)\) | = | \((11,a-4)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 14641 \) | = | \(11^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{4719104528}{14641} a^{2} - \frac{657318240}{14641} a + \frac{42021732273}{14641} \) | ||
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: | \(0 \le r \le 2\) | |
Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
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 generators: | $\left(41 a^{2} + 78 a - 142 : -21 a^{2} - 39 a + 74 : 1\right)$ | $\left(\frac{89}{4} a^{2} + \frac{169}{4} a - \frac{311}{4} : -\frac{93}{8} a^{2} - \frac{169}{8} a + \frac{335}{8} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0 \le r \le 2\) | ||
Regulator*: | \( 1 \) | ||
Period: | \( 176.98219351452631129194285824160749501 \) | ||
Tamagawa product: | \( 4 \) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 4.0006831826651691982707980942917078970 \) | ||
Analytic order of Ш*: | \( 4 \) (rounded) |
* Conditional on BSD: assuming rank = analytic rank.
Note: We expect that the nontriviality of Ш explains the discrepancy between the upper bound on the rank and the analytic rank. The application of further descents should suffice to establish the weak BSD conjecture for this curve.
Local data at primes of bad reduction
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((11,a-4)\) | \(11\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
11.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.