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Properties

Label 265598a
Number of curves $3$
Conductor $265598$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 265598a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
265598.a3 265598a1 \([1, 1, 0, -78201, 8384473]\) \(11134383337/316\) \(1501032940156\) \([]\) \(864000\) \(1.4380\) \(\Gamma_0(N)\)-optimal
265598.a2 265598a2 \([1, 1, 0, -137036, -5912432]\) \(59914169497/31554496\) \(149887145272217536\) \([]\) \(2592000\) \(1.9873\)  
265598.a1 265598a3 \([1, 1, 0, -8768971, -9998373187]\) \(15698803397448457/20709376\) \(98371694766063616\) \([]\) \(7776000\) \(2.5366\)  

Rank

sage: E.rank()
 

The elliptic curves in class 265598a have rank \(1\).

Complex multiplication

The elliptic curves in class 265598a do not have complex multiplication.

Modular form 265598.2.a.a

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.