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Properties

Label 258570.fo
Number of curves $2$
Conductor $258570$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("fo1")
 
E.isogeny_class()
 

Elliptic curves in class 258570.fo

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
258570.fo1 258570fo2 \([1, -1, 1, -7767272, 8260007851]\) \(420644261295449288721/4302712843796480\) \(530098525068570132480\) \([]\) \(12644352\) \(2.7942\)  
258570.fo2 258570fo1 \([1, -1, 1, -694622, -222654779]\) \(300853103177579121/10625000\) \(1309010625000\) \([]\) \(1806336\) \(1.8212\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 258570.fo have rank \(1\).

Complex multiplication

The elliptic curves in class 258570.fo do not have complex multiplication.

Modular form 258570.2.a.fo

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