Base field 4.4.9792.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 2 x + 7 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([7, 2, -7, -2, 1]))
gp: K = nfinit(Polrev([7, 2, -7, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, 2, -7, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([5,-2,-3,1]),K([0,0,0,0]),K([9299,11345,1299,-1420]),K([1304734,1593168,185398,-199304])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([5,-2,-3,1]),Polrev([0,0,0,0]),Polrev([9299,11345,1299,-1420]),Polrev([1304734,1593168,185398,-199304])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![5,-2,-3,1],K![0,0,0,0],K![9299,11345,1299,-1420],K![1304734,1593168,185398,-199304]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-a-1)\) | = | \((a^3-3a^2-3a+4)\cdot(a^3-3a^2-2a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 36 \) | = | \(4\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-4a^3+8a^2+24a-4)\) | = | \((a^3-3a^2-3a+4)^{5}\cdot(a^3-3a^2-2a+5)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -9216 \) | = | \(-4^{5}\cdot9\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{2240392722766771819}{8} a^{3} - \frac{5522297380891621979}{12} a^{2} + \frac{3630089700724133759}{12} a + \frac{13074436924466614051}{24} \) | ||
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: | \(1\) |
| Generator | $\left(-\frac{3701481773}{87146402} a^{3} + \frac{7202413181}{87146402} a^{2} + \frac{18079850910}{43573201} a + \frac{3768640843}{12449486} : -\frac{2179800700292843}{1150506799204} a^{3} + \frac{814529287533821}{1150506799204} a^{2} + \frac{14184526542859451}{1150506799204} a + \frac{866171028728793}{82179057086} : 1\right)$ |
| Height | \(8.7174787478620607136160700414121271567\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 8.7174787478620607136160700414121271567 \) | ||
| Period: | \( 0.36242626726124335242126188939798237474 \) | ||
| Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.19282313038363 \) | ||
| Analytic order of Ш: | \( 25 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-3a^2-3a+4)\) | \(4\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
| \((a^3-3a^2-2a+5)\) | \(9\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
36.1-d
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.