Properties

Label 258570.ef
Number of curves 8
Conductor 258570
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("258570.ef1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 258570.ef

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
258570.ef1 258570ef7 [1, -1, 1, -10024911009158, 12217124280120623777L] [2] 4459069440  
258570.ef2 258570ef6 [1, -1, 1, -626556946658, 190892678863373777] [2, 2] 2229534720  
258570.ef3 258570ef8 [1, -1, 1, -625420790078, 191619465502646081] [2] 4459069440  
258570.ef4 258570ef4 [1, -1, 1, -123768737618, 16757519007800081] [2] 1486356480  
258570.ef5 258570ef3 [1, -1, 1, -39230827538, 2971343928843281] [2] 1114767360  
258570.ef6 258570ef2 [1, -1, 1, -8452601618, 210391515070481] [2, 2] 743178240  
258570.ef7 258570ef1 [1, -1, 1, -3188238098, -66720369131503] [2] 371589120 \(\Gamma_0(N)\)-optimal
258570.ef8 258570ef5 [1, -1, 1, 22633718062, 1398361438905617] [2] 1486356480  

Rank

sage: E.rank()
 

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

Modular form 258570.2.a.ef

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} - q^{5} + 4q^{7} + q^{8} - q^{10} + 4q^{14} + q^{16} - q^{17} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.