Properties

Label 510.c
Number of curves $4$
Conductor $510$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("510.c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 510.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
510.c1 510d3 [1, 1, 1, -6541, -206341] [2] 768  
510.c2 510d2 [1, 1, 1, -421, -3157] [2, 2] 384  
510.c3 510d1 [1, 1, 1, -101, 299] [4] 192 \(\Gamma_0(N)\)-optimal
510.c4 510d4 [1, 1, 1, 579, -14757] [2] 768  

Rank

sage: E.rank()
 

The elliptic curves in class 510.c have rank \(1\).

Modular form 510.2.a.c

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.