Base field 5.5.70601.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, 3, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,0,0,0]),K([5,-1,-7,0,1]),K([0,0,0,0,0]),K([-84,-4,136,12,-24]),K([-327,-49,555,96,-115])])
gp: E = ellinit([Polrev([0,1,0,0,0]),Polrev([5,-1,-7,0,1]),Polrev([0,0,0,0,0]),Polrev([-84,-4,136,12,-24]),Polrev([-327,-49,555,96,-115])], K);
magma: E := EllipticCurve([K![0,1,0,0,0],K![5,-1,-7,0,1],K![0,0,0,0,0],K![-84,-4,136,12,-24],K![-327,-49,555,96,-115]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^4-2a^3-3a^2+5a)\) | = | \((a^4-2a^3-3a^2+5a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 47 \) | = | \(47\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-5a^4+26a^3-20a^2+2a+83)\) | = | \((a^4-2a^3-3a^2+5a)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -10779215329 \) | = | \(-47^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{167855641818319871333}{10779215329} a^{4} + \frac{310535321180235825494}{10779215329} a^{3} + \frac{575318853593468791079}{10779215329} a^{2} - \frac{824740447065649903509}{10779215329} a + \frac{197474025812370152222}{10779215329} \) | ||
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: | \(0\) |
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);
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Torsion generator: | $\left(-a^{4} + \frac{27}{4} a^{2} + a - 5 : \frac{1}{2} a^{4} - \frac{7}{8} a^{3} - \frac{3}{2} a^{2} + a + \frac{1}{2} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 413.88599846389855149204969882437346889 \) | ||
Tamagawa product: | \( 6 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.33650421 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^4-2a^3-3a^2+5a)\) | \(47\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
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 |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
47.1-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.