Properties

Label 65366t
Number of curves $4$
Conductor $65366$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("t1")
 
E.isogeny_class()
 

Elliptic curves in class 65366t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
65366.q4 65366t1 \([1, -1, 1, -2205916, -1272100833]\) \(-10090256344188054273/107965577101312\) \(-12702042180392255488\) \([4]\) \(1843200\) \(2.4820\) \(\Gamma_0(N)\)-optimal
65366.q3 65366t2 \([1, -1, 1, -35384796, -81007585249]\) \(41647175116728660507393/4693358285056\) \(552168908878553344\) \([2, 2]\) \(3686400\) \(2.8286\)  
65366.q2 65366t3 \([1, -1, 1, -35474956, -80573951713]\) \(41966336340198080824833/442001722607124848\) \(52001060663005631242352\) \([2]\) \(7372800\) \(3.1752\)  
65366.q1 65366t4 \([1, -1, 1, -566156716, -5184910367969]\) \(170586815436843383543017473/2166416\) \(254876675984\) \([2]\) \(7372800\) \(3.1752\)  

Rank

sage: E.rank()
 

The elliptic curves in class 65366t have rank \(1\).

Complex multiplication

The elliptic curves in class 65366t do not have complex multiplication.

Modular form 65366.2.a.t

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.