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Properties

Label 259210.cf
Number of curves $4$
Conductor $259210$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cf1")
 
E.isogeny_class()
 

Elliptic curves in class 259210.cf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
259210.cf1 259210cf4 \([1, 1, 1, -4324426891, -109457791336937]\) \(513516182162686336369/1944885031250\) \(33872651195862646515031250\) \([2]\) \(379551744\) \(4.1133\)  
259210.cf2 259210cf3 \([1, 1, 1, -274270641, -1657212524437]\) \(131010595463836369/7704101562500\) \(134176746085778641601562500\) \([2]\) \(189775872\) \(3.7668\)  
259210.cf3 259210cf2 \([1, 1, 1, -73642101, -26081342701]\) \(2535986675931409/1450751712200\) \(25266689768067035704224200\) \([2]\) \(126517248\) \(3.5640\)  
259210.cf4 259210cf1 \([1, 1, 1, -47721101, 126323768899]\) \(690080604747409/3406760000\) \(59333066651168936360000\) \([2]\) \(63258624\) \(3.2174\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 259210.cf have rank \(0\).

Complex multiplication

The elliptic curves in class 259210.cf do not have complex multiplication.

Modular form 259210.2.a.cf

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.