Base field 4.4.2225.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 2 x + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([4, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0]),K([-4,-5/2,1/2,1/2]),K([-2,0,1,0]),K([-1833,-2471,75,507]),K([-40525,-55760,1595,11504])])
gp: E = ellinit([Polrev([1,1,0,0]),Polrev([-4,-5/2,1/2,1/2]),Polrev([-2,0,1,0]),Polrev([-1833,-2471,75,507]),Polrev([-40525,-55760,1595,11504])], K);
magma: E := EllipticCurve([K![1,1,0,0],K![-4,-5/2,1/2,1/2],K![-2,0,1,0],K![-1833,-2471,75,507],K![-40525,-55760,1595,11504]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((-a)\cdot(-1/2a^3+1/2a^2+5/2a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(4\cdot4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((1024)\) | = | \((-a)^{10}\cdot(-1/2a^3+1/2a^2+5/2a-1)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1099511627776 \) | = | \(4^{10}\cdot4^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{2376401567779}{2048} a^{3} + \frac{2376401567779}{2048} a^{2} + \frac{7129204703337}{2048} a + \frac{735163478721}{512} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(\frac{23}{4} a^{3} - \frac{5}{2} a^{2} - \frac{53}{2} a - \frac{37}{4} : -\frac{9}{2} a^{3} - \frac{3}{8} a^{2} + \frac{189}{8} a + \frac{137}{8} : 1\right)$ | $\left(\frac{19}{4} a^{3} + \frac{7}{4} a^{2} - \frac{43}{2} a - 19 : -\frac{45}{8} a^{3} - \frac{5}{2} a^{2} + 25 a + 20 : 1\right)$ |
| 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: | \( 5.5028012772856951706618579650766906381 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 0.729119711002391 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(4\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
| \((-1/2a^3+1/2a^2+5/2a-1)\) | \(4\) | \(2\) | \(I_{10}\) | Non-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 |
|---|---|
| \(2\) | 2Cs |
| \(5\) | 5B.1.4[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
16.1-b
consists of curves linked by isogenies of
degrees dividing 20.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.