Base field 4.4.9225.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 10 x^{2} + 7 x + 19 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([19, 7, -10, -1, 1]))
gp: K = nfinit(Polrev([19, 7, -10, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, 7, -10, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1/4,-1/2,0,1/4]),K([-17/4,1/2,1,-1/4]),K([-4,0,1,0]),K([15/2,7,0,-5/2]),K([51/4,33/2,-2,-17/4])])
gp: E = ellinit([Polrev([1/4,-1/2,0,1/4]),Polrev([-17/4,1/2,1,-1/4]),Polrev([-4,0,1,0]),Polrev([15/2,7,0,-5/2]),Polrev([51/4,33/2,-2,-17/4])], K);
magma: E := EllipticCurve([K![1/4,-1/2,0,1/4],K![-17/4,1/2,1,-1/4],K![-4,0,1,0],K![15/2,7,0,-5/2],K![51/4,33/2,-2,-17/4]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((1/2a^3-3a-1/2)\) | = | \((1/2a^3-3a-1/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 25 \) | = | \(25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-1/2a^3+3a+1/2)\) | = | \((1/2a^3-3a-1/2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 25 \) | = | \(25\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{540595195}{4} a^{3} - \frac{730667933}{5} a^{2} + \frac{10473423273}{10} a + \frac{24675440989}{20} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(1\) |
Generator | $\left(-\frac{687585}{1737124} a^{3} - \frac{358946}{434281} a^{2} + \frac{1611015}{868562} a + \frac{6642611}{1737124} : \frac{291349114}{286191179} a^{3} + \frac{121293716}{286191179} a^{2} - \frac{1194139064}{286191179} a - \frac{825543680}{286191179} : 1\right)$ |
Height | \(2.9396762830165870433785469721138198773\) |
Torsion structure: | \(\Z/8\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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Torsion generator: | $\left(-\frac{1}{4} a^{3} + \frac{1}{2} a + \frac{7}{4} : a^{3} - a^{2} - 3 a + 1 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 2.9396762830165870433785469721138198773 \) | ||
Period: | \( 1324.9430586788292590262857837530332727 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(8\) | ||
Leading coefficient: | \( 2.53450733678301 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/2a^3-3a-1/2)\) | \(25\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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, 4 and 8.
Its isogeny class
25.1-d
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.