Properties

Label 4.4.6125.1-25.1-c3
Base field 4.4.6125.1
Conductor \((-2a^3-2a^2+12a+7)\)
Conductor norm \( 25 \)
CM yes (\(-35\))
Base change no
Q-curve yes
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 4.4.6125.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 9 x^{2} + 9 x + 11 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 9, -9, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([11, 9, -9, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 9, -9, -1, 1]);
 

Weierstrass equation

\({y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-60a^{3}-73a^{2}+376a+297\right){x}-201a^{3}-245a^{2}+1267a+998\)
sage: E = EllipticCurve([K([0,0,0,0]),K([-1,1,0,0]),K([0,1,0,0]),K([297,376,-73,-60]),K([998,1267,-245,-201])])
 
gp: E = ellinit([Pol(Vecrev([0,0,0,0])),Pol(Vecrev([-1,1,0,0])),Pol(Vecrev([0,1,0,0])),Pol(Vecrev([297,376,-73,-60])),Pol(Vecrev([998,1267,-245,-201]))], K);
 
magma: E := EllipticCurve([K![0,0,0,0],K![-1,1,0,0],K![0,1,0,0],K![297,376,-73,-60],K![998,1267,-245,-201]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^3-2a^2+12a+7)\) = \((3a^3+4a^2-17a-13)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 25 \) = \(5^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-10a^3-10a^2+60a+35)\) = \((3a^3+4a^2-17a-13)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 15625 \) = \(5^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -52756480 a^{3} - 52756480 a^{2} + 316538880 a + 125665280 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[(1+\sqrt{-35})/2]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(0\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 26.0448203856482 \)
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: \( 0.665577015360933 \)
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\((j)_{-}\)
\((3a^3+4a^2-17a-13)\) \(5\) \(2\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(5\) 5B.1.4[2]
\(7\) 7B.6.2

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -35 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -35 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5, 7 and 35.
Its isogeny class 25.1-c consists of curves linked by isogenies of degrees dividing 35.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is a \(\Q\)-curve.