Properties

Label 28830.a
Number of curves 8
Conductor 28830
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("28830.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 28830.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
28830.a1 28830h7 [1, 1, 0, -5125513, 4464229117] [2] 691200  
28830.a2 28830h8 [1, 1, 0, -435833, 14891373] [2] 691200  
28830.a3 28830h6 [1, 1, 0, -320513, 69576117] [2, 2] 345600  
28830.a4 28830h5 [1, 1, 0, -277268, -56310078] [2] 230400  
28830.a5 28830h4 [1, 1, 0, -65848, 5574478] [2] 230400  
28830.a6 28830h2 [1, 1, 0, -17798, -835392] [2, 2] 115200  
28830.a7 28830h3 [1, 1, 0, -12993, 1860213] [2] 172800  
28830.a8 28830h1 [1, 1, 0, 1422, -62748] [2] 57600 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 28830.a have rank \(2\).

Modular form 28830.2.a.a

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.