Properties

Label 42978r
Number of curves 2
Conductor 42978
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("42978.u1")
sage: E.isogeny_class()

Elliptic curves in class 42978r

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
42978.u1 42978r1 [1, 0, 0, -459769306, 3796241996132] 7 11063808 \(\Gamma_0(N)\)-optimal
42978.u2 42978r2 [1, 0, 0, 2127889334, -194909741524108] 1 77446656  

Rank

sage: E.rank()

The elliptic curves in class 42978r have rank \(1\).

Modular form 42978.2.a.u

sage: E.q_eigenform(10)
\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - 2q^{11} + q^{12} - q^{13} + q^{14} - q^{15} + q^{16} + 4q^{17} + q^{18} + q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.