Properties

Label 30345c
Number of curves 6
Conductor 30345
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("30345.c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 30345c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
30345.c4 30345c1 [1, 1, 1, -1128551, 460985348] [4] 294912 \(\Gamma_0(N)\)-optimal
30345.c3 30345c2 [1, 1, 1, -1129996, 459743804] [2, 2] 589824  
30345.c5 30345c3 [1, 1, 1, -515871, 958167654] [2] 1179648  
30345.c2 30345c4 [1, 1, 1, -1767241, -118109962] [2, 2] 1179648  
30345.c6 30345c5 [1, 1, 1, 7002464, -931938586] [2] 2359296  
30345.c1 30345c6 [1, 1, 1, -20732866, -36274177462] [2] 2359296  

Rank

sage: E.rank()
 

The elliptic curves in class 30345c have rank \(1\).

Modular form 30345.2.a.c

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} - q^{7} + 3q^{8} + q^{9} + q^{10} + 4q^{11} + q^{12} - 2q^{13} + q^{14} + q^{15} - q^{16} - q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.