Properties

Label 12342.i
Number of curves 6
Conductor 12342
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("12342.i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12342.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12342.i1 12342l5 [1, 0, 1, -3357027, 2367167284] [2] 163840  
12342.i2 12342l3 [1, 0, 1, -209817, 36973000] [2, 2] 81920  
12342.i3 12342l6 [1, 0, 1, -198927, 40984876] [2] 163840  
12342.i4 12342l2 [1, 0, 1, -13797, 513280] [2, 2] 40960  
12342.i5 12342l1 [1, 0, 1, -4117, -94624] [2] 20480 \(\Gamma_0(N)\)-optimal
12342.i6 12342l4 [1, 0, 1, 27343, 2998136] [2] 81920  

Rank

sage: E.rank()
 

The elliptic curves in class 12342.i have rank \(1\).

Modular form 12342.2.a.i

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} - q^{8} + q^{9} + 2q^{10} + q^{12} + 2q^{13} - 2q^{15} + q^{16} - q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.