Properties

Label 248430cb
Number of curves 8
Conductor 248430
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("248430.cb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 248430cb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
248430.cb7 248430cb1 [1, 1, 0, -213028897, -1190430079691] [2] 74317824 \(\Gamma_0(N)\)-optimal
248430.cb6 248430cb2 [1, 1, 0, -342874977, 432048628341] [2, 2] 148635648  
248430.cb5 248430cb3 [1, 1, 0, -1316058097, 17575334152069] [2] 222953472  
248430.cb4 248430cb4 [1, 1, 0, -4110729977, 101275673419341] [2] 297271296  
248430.cb8 248430cb5 [1, 1, 0, 1347442743, 3429658072989] [2] 297271296  
248430.cb2 248430cb6 [1, 1, 0, -20801085477, 1154709841032441] [2, 2] 445906944  
248430.cb1 248430cb7 [1, 1, 0, -332817158027, 73901943591424551] [2] 891813888  
248430.cb3 248430cb8 [1, 1, 0, -20545451007, 1184474231531739] [2] 891813888  

Rank

sage: E.rank()
 

The elliptic curves in class 248430cb have rank \(1\).

Modular form 248430.2.a.cb

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

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 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.