Properties

Label 350727.c
Number of curves $6$
Conductor $350727$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("350727.c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 350727.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
350727.c1 350727c5 [1, 1, 1, -10672057, 13412493314] [2] 12976128  
350727.c2 350727c3 [1, 1, 1, -734792, 164131616] [2, 2] 6488064  
350727.c3 350727c2 [1, 1, 1, -287787, -57582864] [2, 2] 3244032  
350727.c4 350727c1 [1, 1, 1, -285142, -58724446] [2] 1622016 \(\Gamma_0(N)\)-optimal
350727.c5 350727c4 [1, 1, 1, 116898, -206183196] [2] 6488064  
350727.c6 350727c6 [1, 1, 1, 2050393, 1114436738] [2] 12976128  

Rank

sage: E.rank()
 

The elliptic curves in class 350727.c have rank \(1\).

Modular form 350727.2.a.c

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.