Base field 3.3.316.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 4 x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -4, -1, 1]))
gp: K = nfinit(Polrev([2, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,1,1]),K([0,-1,0]),K([0,1,0]),K([2583572058090078441238362,321836266424514724186808,-608022265892705467026233]),K([2667316476142693080354081387519995775,332268331114001902947589285516929027,-627730820434767876188418741768082419])])
gp: E = ellinit([Polrev([-3,1,1]),Polrev([0,-1,0]),Polrev([0,1,0]),Polrev([2583572058090078441238362,321836266424514724186808,-608022265892705467026233]),Polrev([2667316476142693080354081387519995775,332268331114001902947589285516929027,-627730820434767876188418741768082419])], K);
magma: E := EllipticCurve([K![-3,1,1],K![0,-1,0],K![0,1,0],K![2583572058090078441238362,321836266424514724186808,-608022265892705467026233],K![2667316476142693080354081387519995775,332268331114001902947589285516929027,-627730820434767876188418741768082419]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a+1)\) | = | \((-a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 2 \) | = | \(2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-35a^2+8a+147)\) | = | \((-a+1)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 65536 \) | = | \(2^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{4108077233}{65536} a^{2} - \frac{5810733523}{32768} a + \frac{2294990397}{32768} \) | ||
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: | \(\Z/16\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-100400695540 a^{2} + 53143752806 a + 426616665475 : 460599658978230867 a^{2} - 243803036298866838 a - 1957152682813046819 : 1\right)$ |
| 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: | \( 171.47729308324025778719298537841867839 \) | ||
| Tamagawa product: | \( 16 \) | ||
| Torsion order: | \(16\) | ||
| Leading coefficient: | \( 0.60289696166093556063081581682945896038 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+1)\) | \(2\) | \(16\) | \(I_{16}\) | 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, 8 and 16.
Its isogeny class
2.2-a
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.