Properties

Label 333270eg
Number of curves 8
Conductor 333270
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("333270.eg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 333270eg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
333270.eg7 333270eg1 [1, -1, 1, -2368697, 1403230169] [2] 9732096 \(\Gamma_0(N)\)-optimal
333270.eg6 333270eg2 [1, -1, 1, -2749577, 921950201] [2, 2] 19464192  
333270.eg5 333270eg3 [1, -1, 1, -7010672, -5428185901] [2] 29196288  
333270.eg8 333270eg4 [1, -1, 1, 9152923, 6763697201] [2] 38928384  
333270.eg4 333270eg5 [1, -1, 1, -20746157, -35726285311] [2] 38928384  
333270.eg2 333270eg6 [1, -1, 1, -104515952, -411206159149] [2, 2] 58392576  
333270.eg3 333270eg7 [1, -1, 1, -96898352, -473694855469] [2] 116785152  
333270.eg1 333270eg8 [1, -1, 1, -1672218032, -26319677814061] [2] 116785152  

Rank

sage: E.rank()
 

The elliptic curves in class 333270eg have rank \(1\).

Modular form 333270.2.a.eg

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

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.