Properties

Label 35.a
Number of curves $3$
Conductor $35$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 35.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
35.a1 35a2 \([0, 1, 1, -131, -650]\) \(-250523582464/13671875\) \(-13671875\) \([]\) \(6\) \(0.12746\)  
35.a2 35a3 \([0, 1, 1, -1, 0]\) \(-262144/35\) \(-35\) \([3]\) \(6\) \(-0.97115\)  
35.a3 35a1 \([0, 1, 1, 9, 1]\) \(71991296/42875\) \(-42875\) \([3]\) \(2\) \(-0.42184\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 35.a have rank \(0\).

Complex multiplication

The elliptic curves in class 35.a do not have complex multiplication.

Modular form 35.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2q^{4} - q^{5} + q^{7} - 2q^{9} - 3q^{11} - 2q^{12} + 5q^{13} - q^{15} + 4q^{16} + 3q^{17} + 2q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.