Properties

Label 1344a
Number of curves 6
Conductor 1344
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("1344.g1")
sage: E.isogeny_class()

Elliptic curves in class 1344a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
1344.g6 1344a1 [0, -1, 0, 63, -63] 2 256 \(\Gamma_0(N)\)-optimal
1344.g5 1344a2 [0, -1, 0, -257, -255] 4 512  
1344.g2 1344a3 [0, -1, 0, -3137, -66495] 4 1024  
1344.g3 1344a4 [0, -1, 0, -2497, 48577] 2 1024  
1344.g1 1344a5 [0, -1, 0, -50177, -4309503] 2 2048  
1344.g4 1344a6 [0, -1, 0, -2177, -108927] 2 2048  

Rank

sage: E.rank()

The elliptic curves in class 1344a have rank \(1\).

Modular form 1344.2.a.g

sage: E.q_eigenform(10)
\( q - q^{3} + 2q^{5} - q^{7} + q^{9} - 4q^{11} + 2q^{13} - 2q^{15} - 6q^{17} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.