Properties

Label 2.2.105.1-35.1-a2
Base field \(\Q(\sqrt{105}) \)
Conductor \((35,a+17)\)
Conductor norm \( 35 \)
CM no
Base change yes: 175.b2,2205.e2
Q-curve yes
Torsion order \( 1 \)
Rank \( 2 \)

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Show commands: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{105}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 26 \); class number \(2\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-26, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([-26, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-26, -1, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-875a-4035\right){x}+36705a+169711\)
sage: E = EllipticCurve([K([0,0]),K([1,-1]),K([1,0]),K([-4035,-875]),K([169711,36705])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,-1])),Pol(Vecrev([1,0])),Pol(Vecrev([-4035,-875])),Pol(Vecrev([169711,36705]))], K);
 
magma: E := EllipticCurve([K![0,0],K![1,-1],K![1,0],K![-4035,-875],K![169711,36705]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((35,a+17)\) = \((2a-11)\cdot(7,a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 35 \) = \(5\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-35)\) = \((2a-11)^{2}\cdot(7,a+3)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1225 \) = \(5^{2}\cdot7^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{262144}{35} \)
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 $\left(\frac{141}{16} a + \frac{311}{8} : \frac{3057}{64} a + \frac{7051}{32} : 1\right)$ $\left(-5 a - 25 : -81 a - 375 : 1\right)$
Heights \(0.543369538023551\) \(0.0925733338021421\)
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: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 0.0417317074901272 \)
Period: \( 40.0208210182756 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 2.60781921864617 \)
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\((j)_{-}\)
\((2a-11)\) \(5\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((7,a+3)\) \(7\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 35.1-a consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base change of 175.b2, 2205.e2, defined over \(\Q\), so it is also a \(\Q\)-curve.