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Properties

Label 4.4.6125.1-25.1-b2
Base field 4.4.6125.1
Conductor norm \( 25 \)
CM yes (\(-28\))
Base change no
Q-curve yes
Torsion order \( 2 \)
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(Polrev([11, 9, -9, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 9, -9, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{3}+3a^{2}-11a-11\right){x}{y}+\left(a^{3}+2a^{2}-6a-8\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-48a^{3}-78a^{2}+296a+242\right){x}+110a^{3}+62a^{2}-507a-369\)
sage: E = EllipticCurve([K([-11,-11,3,2]),K([-3,0,1,0]),K([-8,-6,2,1]),K([242,296,-78,-48]),K([-369,-507,62,110])])
 
gp: E = ellinit([Polrev([-11,-11,3,2]),Polrev([-3,0,1,0]),Polrev([-8,-6,2,1]),Polrev([242,296,-78,-48]),Polrev([-369,-507,62,110])], K);
 
magma: E := EllipticCurve([K![-11,-11,3,2],K![-3,0,1,0],K![-8,-6,2,1],K![242,296,-78,-48],K![-369,-507,62,110]]);
 

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: \( 16581375 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[\sqrt{-7}]\) (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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{9}{4} a^{3} + 2 a^{2} - \frac{49}{4} a - \frac{17}{2} : -\frac{39}{8} a^{3} - \frac{15}{2} a^{2} + 29 a + 26 : 1\right)$
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: \( 306.13895805951269022674653388467207885 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 1.95585005937722 \)
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
\(5\) 5Nn.2.2[2]

For all other primes \(p\), the image is a Borel subgroup if \(p\in \{ 2, 7\}\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 7 and 14.
Its isogeny class 25.1-b consists of curves linked by isogenies of degrees dividing 14.

Base change

This elliptic curve is a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.