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Properties

Label 51870.m
Number of curves $2$
Conductor $51870$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 51870.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51870.m1 51870q1 \([1, 1, 0, -6247, -192671]\) \(26967882743214841/58820580\) \(58820580\) \([2]\) \(51200\) \(0.73740\) \(\Gamma_0(N)\)-optimal
51870.m2 51870q2 \([1, 1, 0, -6177, -197109]\) \(-26071520989028761/1260882154350\) \(-1260882154350\) \([2]\) \(102400\) \(1.0840\)  

Rank

sage: E.rank()
 

The elliptic curves in class 51870.m have rank \(1\).

Complex multiplication

The elliptic curves in class 51870.m do not have complex multiplication.

Modular form 51870.2.a.m

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.