Base field 5.5.89417.1
Generator \(a\), with minimal polynomial \( x^{5} - 6 x^{3} - x^{2} + 8 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 8, -1, -6, 0, 1]))
gp: K = nfinit(Polrev([3, 8, -1, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 8, -1, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,1,-4,0,1]),K([3,4,-4,-1,1]),K([4,0,-5,0,1]),K([88,207,-120,-67,19]),K([587,1448,-321,-539,-36])])
gp: E = ellinit([Polrev([3,1,-4,0,1]),Polrev([3,4,-4,-1,1]),Polrev([4,0,-5,0,1]),Polrev([88,207,-120,-67,19]),Polrev([587,1448,-321,-539,-36])], K);
magma: E := EllipticCurve([K![3,1,-4,0,1],K![3,4,-4,-1,1],K![4,0,-5,0,1],K![88,207,-120,-67,19],K![587,1448,-321,-539,-36]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-a^2-3a+2)\) | = | \((a^3-a^2-3a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 11 \) | = | \(11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1383a^4+977a^3+6373a^2-2399a-4820)\) | = | \((a^3-a^2-3a+2)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 45949729863572161 \) | = | \(11^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{476417640375476377168321}{45949729863572161} a^{4} - \frac{921552595192807794564289}{45949729863572161} a^{3} - \frac{1018805365320562990606080}{45949729863572161} a^{2} + \frac{1637467232063013399273925}{45949729863572161} a + \frac{751306184458928265884514}{45949729863572161} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(0 \le r \le 1\) |
| 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: | $\left(-\frac{7}{2} a^{4} + \frac{7}{4} a^{3} + 13 a^{2} - 9 a - 3 : -\frac{3}{2} a^{4} + \frac{73}{8} a^{3} + \frac{115}{8} a^{2} - 26 a - \frac{65}{4} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0 \le r \le 1\) | ||
| Regulator: | not available | ||
| Period: | \( 292.63955810705728365109250672219830032 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.87789849 \) | ||
| Analytic order of Ш: | not available | ||
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-3a+2)\) | \(11\) | \(2\) | \(I_{16}\) | 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 and 4.
Its isogeny class
11.1-b
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.