Base field 4.4.13888.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 6 x + 9 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, 6, -7, -2, 1]))
gp: K = nfinit(Polrev([9, 6, -7, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 6, -7, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1,0]),K([3,1,-1,0]),K([-3,-4/3,1/3,1/3]),K([-237,208/3,200/3,-37/3]),K([-1362,1838/3,925/3,-293/3])])
gp: E = ellinit([Polrev([-3,0,1,0]),Polrev([3,1,-1,0]),Polrev([-3,-4/3,1/3,1/3]),Polrev([-237,208/3,200/3,-37/3]),Polrev([-1362,1838/3,925/3,-293/3])], K);
magma: E := EllipticCurve([K![-3,0,1,0],K![3,1,-1,0],K![-3,-4/3,1/3,1/3],K![-237,208/3,200/3,-37/3],K![-1362,1838/3,925/3,-293/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/3a^3-5/3a^2-1/3a+4)\) | = | \((1/3a^3-2/3a^2-4/3a+1)\cdot(-a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 28 \) | = | \(4\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((4/3a^3-20/3a^2-4/3a+16)\) | = | \((1/3a^3-2/3a^2-4/3a+1)^{5}\cdot(-a+1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -7168 \) | = | \(-4^{5}\cdot7\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{5936809291233745480211}{56} a^{3} - \frac{17240964829393127426231}{56} a^{2} - \frac{25970465306010387480099}{56} a + \frac{29550107113814174736753}{28} \) | ||
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 | $\left(-\frac{1633}{98} a^{3} + \frac{3153}{98} a^{2} + \frac{3513}{49} a + \frac{809}{49} : \frac{167077}{4116} a^{3} + \frac{223885}{1029} a^{2} - \frac{913015}{1029} a - \frac{1316727}{1372} : 1\right)$ |
| Height | \(3.9509304655093517127075385164850877798\) |
| 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: | \( 3.9509304655093517127075385164850877798 \) | ||
| Period: | \( 0.96331394842300898980927001016208368094 \) | ||
| Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.22958992221777 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/3a^3-2/3a^2-4/3a+1)\) | \(4\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
| \((-a+1)\) | \(7\) | \(1\) | \(I_{1}\) | 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
28.3-a
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.