Properties

Label 61347.i
Number of curves $6$
Conductor $61347$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("61347.i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 61347.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
61347.i1 61347z4 [1, 0, 0, -140362362, -640076749443] [2] 5160960  
61347.i2 61347z6 [1, 0, 0, -61531467, 179889890478] [2] 10321920  
61347.i3 61347z3 [1, 0, 0, -9693252, -7774815465] [2, 2] 5160960  
61347.i4 61347z2 [1, 0, 0, -8773047, -10000791360] [2, 2] 2580480  
61347.i5 61347z1 [1, 0, 0, -491202, -190117773] [4] 1290240 \(\Gamma_0(N)\)-optimal
61347.i6 61347z5 [1, 0, 0, 27421683, -52973383308] [2] 10321920  

Rank

sage: E.rank()
 

The elliptic curves in class 61347.i have rank \(1\).

Modular form 61347.2.a.i

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.