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Properties

Label 5010.g
Number of curves $4$
Conductor $5010$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("g1")
 
E.isogeny_class()
 

Elliptic curves in class 5010.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5010.g1 5010g4 \([1, 0, 0, -10701, 425181]\) \(135518735544698449/782812500\) \(782812500\) \([2]\) \(6144\) \(0.89697\)  
5010.g2 5010g3 \([1, 0, 0, -2181, -31755]\) \(1147369112034769/233338896300\) \(233338896300\) \([2]\) \(6144\) \(0.89697\)  
5010.g3 5010g2 \([1, 0, 0, -681, 6345]\) \(34930508298769/2510010000\) \(2510010000\) \([2, 2]\) \(3072\) \(0.55040\)  
5010.g4 5010g1 \([1, 0, 0, 39, 441]\) \(6549699311/86572800\) \(-86572800\) \([4]\) \(1536\) \(0.20382\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 5010.g have rank \(1\).

Complex multiplication

The elliptic curves in class 5010.g do not have complex multiplication.

Modular form 5010.2.a.g

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.