Properties

Label 67830.s
Number of curves 8
Conductor 67830
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("67830.s1")
sage: E.isogeny_class()

Elliptic curves in class 67830.s

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
67830.s1 67830p8 [1, 0, 1, -2903932529, 60196943047556] 2 59719680  
67830.s2 67830p5 [1, 0, 1, -2903465114, 60217301097812] 6 19906560  
67830.s3 67830p6 [1, 0, 1, -217399409, 541937811332] 4 29859840  
67830.s4 67830p2 [1, 0, 1, -181466594, 940883727476] 12 9953280  
67830.s5 67830p4 [1, 0, 1, -180445994, 951990304916] 6 19906560  
67830.s6 67830p3 [1, 0, 1, -111231089, -445724835964] 2 14929920  
67830.s7 67830p1 [1, 0, 1, -11405474, 14526794612] 6 4976640 \(\Gamma_0(N)\)-optimal
67830.s8 67830p7 [1, 0, 1, 770440591, 4097766675332] 2 59719680  

Rank

sage: E.rank()

The elliptic curves in class 67830.s have rank \(1\).

Modular form 67830.2.a.s

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 6 & 12 & 4 & 12 & 4 \\ 3 & 1 & 6 & 2 & 4 & 12 & 4 & 12 \\ 2 & 6 & 1 & 3 & 6 & 2 & 6 & 2 \\ 6 & 2 & 3 & 1 & 2 & 6 & 2 & 6 \\ 12 & 4 & 6 & 2 & 1 & 12 & 4 & 3 \\ 4 & 12 & 2 & 6 & 12 & 1 & 3 & 4 \\ 12 & 4 & 6 & 2 & 4 & 3 & 1 & 12 \\ 4 & 12 & 2 & 6 & 3 & 4 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.