Properties

Label 507.c
Number of curves $2$
Conductor $507$
CM no
Rank $1$
Graph

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

Elliptic curves in class 507.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
507.c1 507a2 \([1, 1, 0, -12678, -3060351]\) \(-276301129/4782969\) \(-3901614750890649\) \([]\) \(2184\) \(1.6728\)  
507.c2 507a1 \([1, 1, 0, -1693, 26434]\) \(-658489/9\) \(-7341576489\) \([]\) \(312\) \(0.69989\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 507.c have rank \(1\).

Complex multiplication

The elliptic curves in class 507.c do not have complex multiplication.

Modular form 507.2.a.c

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.