Properties

Label 1785.e
Number of curves $6$
Conductor $1785$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1785.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1785.e1 1785o5 \([1, 0, 0, -71740, -7387525]\) \(40832710302042509761/91556816413125\) \(91556816413125\) \([2]\) \(8192\) \(1.5612\)  
1785.e2 1785o3 \([1, 0, 0, -6115, -24400]\) \(25288177725059761/14387797265625\) \(14387797265625\) \([2, 2]\) \(4096\) \(1.2146\)  
1785.e3 1785o2 \([1, 0, 0, -3910, 93347]\) \(6610905152742241/35128130625\) \(35128130625\) \([2, 4]\) \(2048\) \(0.86806\)  
1785.e4 1785o1 \([1, 0, 0, -3905, 93600]\) \(6585576176607121/187425\) \(187425\) \([4]\) \(1024\) \(0.52149\) \(\Gamma_0(N)\)-optimal
1785.e5 1785o4 \([1, 0, 0, -1785, 194922]\) \(-629004249876241/16074715228425\) \(-16074715228425\) \([8]\) \(4096\) \(1.2146\)  
1785.e6 1785o6 \([1, 0, 0, 24230, -188263]\) \(1573196002879828319/926055908203125\) \(-926055908203125\) \([2]\) \(8192\) \(1.5612\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1785.e have rank \(1\).

Complex multiplication

The elliptic curves in class 1785.e do not have complex multiplication.

Modular form 1785.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} + q^{7} + 3 q^{8} + q^{9} - q^{10} - 4 q^{11} - q^{12} - 2 q^{13} - q^{14} + q^{15} - q^{16} + q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.