Properties

Label 72897a
Number of curves 4
Conductor 72897
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("72897.a1")
sage: E.isogeny_class()

Elliptic curves in class 72897a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
72897.a3 72897a1 [1, 1, 0, -14404, 653827] 2 154560 \(\Gamma_0(N)\)-optimal
72897.a2 72897a2 [1, 1, 0, -25449, -501480] 4 309120  
72897.a4 72897a3 [1, 1, 0, 96046, -3781845] 2 618240  
72897.a1 72897a4 [1, 1, 0, -323664, -70939863] 2 618240  

Rank

sage: E.rank()

The elliptic curves in class 72897a have rank \(0\).

Modular form 72897.2.a.a

sage: E.q_eigenform(10)
\( q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} + 4q^{7} - 3q^{8} + q^{9} + 2q^{10} - q^{11} + q^{12} + 2q^{13} + 4q^{14} - 2q^{15} - q^{16} - 2q^{17} + q^{18} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.