Properties

Label 2730.bd
Number of curves 8
Conductor 2730
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("2730.bd1")
sage: E.isogeny_class()

Elliptic curves in class 2730.bd

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
2730.bd1 2730bd7 [1, 0, 0, -40190455, -98072518885] 2 110592  
2730.bd2 2730bd6 [1, 0, 0, -2511905, -1532538075] 4 55296  
2730.bd3 2730bd8 [1, 0, 0, -2481035, -1572033153] 2 110592  
2730.bd4 2730bd4 [1, 0, 0, -496405, -134437975] 6 36864  
2730.bd5 2730bd3 [1, 0, 0, -158925, -23336703] 4 27648  
2730.bd6 2730bd2 [1, 0, 0, -41405, -576975] 12 18432  
2730.bd7 2730bd1 [1, 0, 0, -25725, 1577457] 12 9216 \(\Gamma_0(N)\)-optimal
2730.bd8 2730bd5 [1, 0, 0, 162715, -4536903] 6 36864  

Rank

sage: E.rank()

The elliptic curves in class 2730.bd have rank \(0\).

Modular form 2730.2.a.bd

sage: E.q_eigenform(10)
\( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} + q^{12} + q^{13} + q^{14} + q^{15} + q^{16} + 6q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.