Properties

Label 12870.c
Number of curves 8
Conductor 12870
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("12870.c1")
sage: E.isogeny_class()

Elliptic curves in class 12870.c

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
12870.c1 12870p7 [1, -1, 0, -9423000, 6261910740] 6 1327104  
12870.c2 12870p6 [1, -1, 0, -8225100, 9078652800] 12 663552  
12870.c3 12870p3 [1, -1, 0, -8224380, 9080321616] 6 331776  
12870.c4 12870p8 [1, -1, 0, -7038720, 11788581996] 6 1327104  
12870.c5 12870p4 [1, -1, 0, -4247100, -3367494000] 2 442368  
12870.c6 12870p2 [1, -1, 0, -287100, -43470000] 4 221184  
12870.c7 12870p1 [1, -1, 0, -102780, 12157776] 2 110592 \(\Gamma_0(N)\)-optimal
12870.c8 12870p5 [1, -1, 0, 723780, -281835504] 2 442368  

Rank

sage: E.rank()

The elliptic curves in class 12870.c have rank \(0\).

Modular form None

sage: E.q_eigenform(10)
\( q - q^{2} + q^{4} - q^{5} - 4q^{7} - q^{8} + q^{10} + q^{11} + q^{13} + 4q^{14} + q^{16} + 6q^{17} - 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.