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Properties

Label 75810.df
Number of curves $2$
Conductor $75810$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("df1")
 
E.isogeny_class()
 

Elliptic curves in class 75810.df

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75810.df1 75810dc1 \([1, 0, 0, -6591, -206379]\) \(4616586342451/3307500\) \(22686142500\) \([2]\) \(138240\) \(0.92238\) \(\Gamma_0(N)\)-optimal
75810.df2 75810dc2 \([1, 0, 0, -5261, -291765]\) \(-2347864201171/3986718750\) \(-27344903906250\) \([2]\) \(276480\) \(1.2690\)  

Rank

sage: E.rank()
 

The elliptic curves in class 75810.df have rank \(0\).

Complex multiplication

The elliptic curves in class 75810.df do not have complex multiplication.

Modular form 75810.2.a.df

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