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([1,0]),K([-1,0]),K([1,0]),K([-847240411,109626655]),K([-12982021956321,1679777810708])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-847240411,109626655]),Polrev([-12982021956321,1679777810708])], K);
magma: E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![-847240411,109626655],K![-12982021956321,1679777810708]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((11a+74)\cdot(11a-85)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 4 \) | = | \(2\cdot2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-2624a-17600)\) | = | \((11a+74)^{6}\cdot(11a-85)^{15}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -2097152 \) | = | \(-2^{6}\cdot2^{15}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{239886134047017}{32768} a + \frac{463484975124303}{8192} \) | ||
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: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{6359}{4} a - 12286 : -\frac{6359}{8} a + \frac{12285}{2} : 1\right)$ |
| 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: | \( 2.6362523835993780374882083537562652817 \) | ||
| Tamagawa product: | \( 90 \) = \(( 2 \cdot 3 )\cdot( 3 \cdot 5 )\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 4.1029512843084866138662251432427598297 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((11a+74)\) | \(2\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((11a-85)\) | \(2\) | \(15\) | \(I_{15}\) | Split multiplicative | \(-1\) | \(1\) | \(15\) | \(15\) |
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 |
| \(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
4.1-c
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.