Properties

Label 413712bb
Number of curves $6$
Conductor $413712$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("413712.bb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 413712bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
413712.bb4 413712bb1 [0, 0, 0, -13117611, 18286461466] [2] 11010048 \(\Gamma_0(N)\)-optimal
413712.bb3 413712bb2 [0, 0, 0, -13239291, 17929914730] [2, 2] 22020096  
413712.bb5 413712bb3 [0, 0, 0, 5377749, 64349642266] [2] 44040192  
413712.bb2 413712bb4 [0, 0, 0, -33803211, -51308803910] [2, 2] 44040192  
413712.bb6 413712bb5 [0, 0, 0, 94325829, -347466266966] [2] 88080384  
413712.bb1 413712bb6 [0, 0, 0, -490954971, -4186429333814] [2] 88080384  

Rank

sage: E.rank()
 

The elliptic curves in class 413712bb have rank \(1\).

Modular form 413712.2.a.bb

sage: E.q_eigenform(10)
 
\( q - 2q^{5} - 4q^{11} - q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.