Properties

Label 407330u
Number of curves 4
Conductor 407330
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("407330.u1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 407330u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
407330.u3 407330u1 [1, 0, 0, -29635, -38214975] [u'2'] 7299072 \(\Gamma_0(N)\)-optimal*
407330.u2 407330u2 [1, 0, 0, -1891715, -992717183] [u'2'] 14598144 \(\Gamma_0(N)\)-optimal*
407330.u4 407330u3 [1, 0, 0, 266605, 1029374737] [u'2'] 21897216 \(\Gamma_0(N)\)-optimal*
407330.u1 407330u4 [1, 0, 0, -13815375, 19203578125] [u'2'] 43794432 \(\Gamma_0(N)\)-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 407330u1.

Rank

sage: E.rank()
 

The elliptic curves in class 407330u have rank \(1\).

Modular form 407330.2.a.u

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

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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.