Properties

Label 3.3.733.1-175.2-c4
Base field 3.3.733.1
Conductor norm \( 175 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 2 \)

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Base field 3.3.733.1

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 7 x + 8 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -7, -1, 1]))
 
gp: K = nfinit(Polrev([8, -7, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(24a^{2}+52a-250\right){x}-245a^{2}-473a+1667\)
sage: E = EllipticCurve([K([1,1,0]),K([-4,1,1]),K([-4,1,1]),K([-250,52,24]),K([1667,-473,-245])])
 
gp: E = ellinit([Polrev([1,1,0]),Polrev([-4,1,1]),Polrev([-4,1,1]),Polrev([-250,52,24]),Polrev([1667,-473,-245])], K);
 
magma: E := EllipticCurve([K![1,1,0],K![-4,1,1],K![-4,1,1],K![-250,52,24],K![1667,-473,-245]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^2-2a+19)\) = \((-a+3)^{2}\cdot(a^2+2a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 175 \) = \(5^{2}\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-8a^2+7a-19)\) = \((-a+3)^{6}\cdot(a^2+2a-3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 109375 \) = \(5^{6}\cdot7\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{84096891433574753384471}{7} a^{2} - \frac{14985568201611275023622}{7} a + \frac{571022332294137577763767}{7} \)
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: \(2\)
Generators $\left(\frac{381381}{400} a^{2} + \frac{580339}{400} a - \frac{1204971}{400} : \frac{613146083}{8000} a^{2} + \frac{930648261}{8000} a - \frac{15587297}{64} : 1\right)$ $\left(\frac{3113}{256} a^{2} + \frac{4623}{256} a - \frac{7895}{256} : -\frac{625383}{4096} a^{2} - \frac{971745}{4096} a + \frac{1983321}{4096} : 1\right)$
Heights \(7.5562684873421855603414737336264051769\) \(2.0774575032097665927940433141672827718\)
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: $\left(-\frac{5}{4} a^{2} + \frac{9}{2} a + \frac{15}{4} : -\frac{3}{2} a^{2} - \frac{1}{4} a - \frac{39}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 1.1913307226662626356620933618202261073 \)
Period: \( 18.100453437762898526170094753536346687 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 3.5841181921351919282727124687104657817 \)
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\((j)_{-}\)
\((-a+3)\) \(5\) \(2\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)
\((a^2+2a-3)\) \(7\) \(1\) \(I_{1}\) 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 175.2-c 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.