Base field \(\Q(\sqrt{209}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 52 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-52, -1, 1]))
gp: K = nfinit(Polrev([-52, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-52, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([1,0]),K([-41686761846,-6195627759]),K([8697866248261405,1292706345403712])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,-1]),Polrev([1,0]),Polrev([-41686761846,-6195627759]),Polrev([8697866248261405,1292706345403712])], K);
magma: E := EllipticCurve([K![0,0],K![-1,-1],K![1,0],K![-41686761846,-6195627759],K![8697866248261405,1292706345403712]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((70a+471)\) | = | \((70a+471)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 11 \) | = | \(11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-161051)\) | = | \((70a+471)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 25937424601 \) | = | \(11^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( -\frac{122023936}{161051} \) | ||
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 | $\left(-12343 a - \frac{332203}{4} : \frac{23554553}{2} a + \frac{633939335}{8} : 1\right)$ |
Height | \(1.1225675167272302521928779174776140001\) |
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: | \( 1.1225675167272302521928779174776140001 \) | ||
Period: | \( 8.5125836874235053042001518168328721296 \) | ||
Tamagawa product: | \( 10 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 13.219977566399926460933276101116182590 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((70a+471)\) | \(11\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(5\) | 5Cs.4.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
11.1-b
consists of curves linked by isogenies of
degrees dividing 25.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 121.d2 |
\(\Q\) | 3971.b2 |