Base field \(\Q(\sqrt{105}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 26 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-26, -1, 1]))
gp: K = nfinit(Polrev([-26, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-26, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,1]),K([-237,40]),K([1141,-204])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([0,1]),Polrev([-237,40]),Polrev([1141,-204])], K);
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,1],K![-237,40],K![1141,-204]]);
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2a-11)\) | = | \((2a-11)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 5 \) | = | \(5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-215a-730)\) | = | \((2,a)^{12}\cdot(2a-11)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( -512000 \) | = | \(-2^{12}\cdot5^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
Minimal discriminant: | \((10a-55)\) | = | \((2a-11)^{3}\) |
Minimal discriminant norm: | \( -125 \) | = | \(-5^{3}\) |
j-invariant: | \( -\frac{1145960298}{25} a + \frac{6444279999}{25} \) | ||
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 | $\left(-\frac{15}{16} a + \frac{35}{8} : -\frac{149}{64} a + \frac{393}{32} : 1\right)$ |
Height | \(0.34462425971263075326913527159199330199\) |
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: | $\left(-a + 4 : -2 a + 13 : 1\right)$ |
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: | \(1\) | ||
Regulator: | \( 0.34462425971263075326913527159199330199 \) | ||
Period: | \( 33.762540382564968316276389652354882852 \) | ||
Tamagawa product: | \( 3 \) = \(1\cdot3\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.7032467635170146272480656800952857667 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
\((2a-11)\) | \(5\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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 |
\(3\) | 3Nn |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
5.1-d
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.