Base field 4.4.4913.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 6 x^{2} + x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 1, -6, -1, 1]))
gp: K = nfinit(Polrev([1, 1, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1/2,3,1,-1/2]),K([3,2,-1,0]),K([1,1,0,0]),K([-1504,-4571,-373,731]),K([-70683/2,-107728,-8824,34473/2])])
gp: E = ellinit([Polrev([-1/2,3,1,-1/2]),Polrev([3,2,-1,0]),Polrev([1,1,0,0]),Polrev([-1504,-4571,-373,731]),Polrev([-70683/2,-107728,-8824,34473/2])], K);
magma: E := EllipticCurve([K![-1/2,3,1,-1/2],K![3,2,-1,0],K![1,1,0,0],K![-1504,-4571,-373,731],K![-70683/2,-107728,-8824,34473/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((1/2a^3-a^2-2a+5/2)\cdot(-a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(4\cdot4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-192a^3+1216a+256)\) | = | \((1/2a^3-a^2-2a+5/2)^{6}\cdot(-a-1)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 68719476736 \) | = | \(4^{6}\cdot4^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{203862548967}{4096} a^{3} - \frac{203862548967}{4096} a^{2} - \frac{1019312744835}{4096} a + \frac{130566616997}{1024} \) | ||
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: | \(0\) | |
| 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);
| ||
| Torsion generators: | $\left(9 a^{3} - 5 a^{2} - 56 a - 16 : -\frac{9}{2} a^{3} + 2 a^{2} + 28 a + \frac{21}{2} : 1\right)$ | $\left(\frac{41}{8} a^{3} - \frac{9}{4} a^{2} - 33 a - \frac{107}{8} : -\frac{7}{2} a^{3} + \frac{19}{8} a^{2} + \frac{177}{8} a + \frac{41}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 26.035750901914682695413858613181622022 \) | ||
| Tamagawa product: | \( 72 \) = \(( 2 \cdot 3 )\cdot( 2^{2} \cdot 3 )\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.67151100189927 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3-a^2-2a+5/2)\) | \(4\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((-a-1)\) | \(4\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
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 |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
16.1-b
consists of curves linked by isogenies of
degrees dividing 24.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q(\sqrt{17}) \) | 2.2.17.1-4.1-a10 |
| \(\Q(\sqrt{17}) \) | a curve with conductor norm 1156 (not in the database) |