Properties

Label 34.a
Number of curves $4$
Conductor $34$
CM no
Rank $0$
Graph

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

Elliptic curves in class 34.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
34.a1 34a4 \([1, 0, 0, -113, -329]\) \(159661140625/48275138\) \(48275138\) \([2]\) \(12\) \(0.17949\)  
34.a2 34a3 \([1, 0, 0, -103, -411]\) \(120920208625/19652\) \(19652\) \([2]\) \(6\) \(-0.16709\)  
34.a3 34a2 \([1, 0, 0, -43, 105]\) \(8805624625/2312\) \(2312\) \([6]\) \(4\) \(-0.36982\)  
34.a4 34a1 \([1, 0, 0, -3, 1]\) \(3048625/1088\) \(1088\) \([6]\) \(2\) \(-0.71639\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 34.a have rank \(0\).

Complex multiplication

The elliptic curves in class 34.a do not have complex multiplication.

Modular form 34.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} - 4 q^{7} + q^{8} + q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} - 4 q^{14} + q^{16} - q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) Copy content 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.