Properties

Label 51.a
Number of curves $2$
Conductor $51$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 51.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51.a1 51a2 \([0, 1, 1, -59, -196]\) \(-23100424192/14739\) \(-14739\) \([]\) \(6\) \(-0.25922\)  
51.a2 51a1 \([0, 1, 1, 1, -1]\) \(32768/459\) \(-459\) \([3]\) \(2\) \(-0.80853\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 51.a have rank \(0\).

Complex multiplication

The elliptic curves in class 51.a do not have complex multiplication.

Modular form 51.2.a.a

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.