Properties

Label 786e
Number of curves $4$
Conductor $786$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 786e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
786.d4 786e1 \([1, 1, 0, -29, -3]\) \(2845178713/1609728\) \(1609728\) \([2]\) \(144\) \(-0.11906\) \(\Gamma_0(N)\)-optimal
786.d2 786e2 \([1, 1, 0, -349, 2365]\) \(4722184089433/9884736\) \(9884736\) \([2, 2]\) \(288\) \(0.22752\)  
786.d1 786e3 \([1, 1, 0, -5589, 158517]\) \(19312898130234073/84888\) \(84888\) \([2]\) \(576\) \(0.57409\)  
786.d3 786e4 \([1, 1, 0, -229, 4165]\) \(-1337180541913/7067998104\) \(-7067998104\) \([2]\) \(576\) \(0.57409\)  

Rank

sage: E.rank()
 

The elliptic curves in class 786e have rank \(0\).

Complex multiplication

The elliptic curves in class 786e do not have complex multiplication.

Modular form 786.2.a.e

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