Base field \(\Q(\sqrt{21}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 5 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-5, -1, 1]))
gp: K = nfinit(Polrev([-5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([-1,-1]),K([1,0]),K([-3203,626]),K([61171,-26113])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,-1]),Polrev([1,0]),Polrev([-3203,626]),Polrev([61171,-26113])], K);
magma: E := EllipticCurve([K![1,0],K![-1,-1],K![1,0],K![-3203,626],K![61171,-26113]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-20a+10)\) | = | \((-a+2)\cdot(2)\cdot(-a)\cdot(-a+1)\cdot(a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 2100 \) | = | \(3\cdot4\cdot5\cdot5\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-8266860a+33067440)\) | = | \((-a+2)^{20}\cdot(2)^{2}\cdot(-a)\cdot(-a+1)\cdot(a+3)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 478386819817200 \) | = | \(3^{20}\cdot4^{2}\cdot5\cdot5\cdot7^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( -\frac{28534491856684081127}{57868020} a + \frac{8849775613116843067}{6429780} \) | ||
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 | $\left(-15 a + 31 : -33 a + 146 : 1\right)$ |
Height | \(0.17389643308748950810839595684666108885\) |
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: | $\left(-15 a + \frac{43}{4} : \frac{15}{2} a - \frac{47}{8} : 1\right)$ |
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: | \( 0.17389643308748950810839595684666108885 \) | ||
Period: | \( 2.1184335229648479743479163549562120313 \) | ||
Tamagawa product: | \( 120 \) = \(( 2^{2} \cdot 5 )\cdot2\cdot1\cdot1\cdot3\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 4.8233315658970428705222335647830861844 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a+2)\) | \(3\) | \(20\) | \(I_{20}\) | Split multiplicative | \(-1\) | \(1\) | \(20\) | \(20\) |
\((2)\) | \(4\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
\((-a)\) | \(5\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((-a+1)\) | \(5\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
\((a+3)\) | \(7\) | \(3\) | \(I_{3}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
2100.1-h
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.