Properties

Label 30030.p
Number of curves 8
Conductor 30030
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("30030.p1")
sage: E.isogeny_class()

Elliptic curves in class 30030.p

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
30030.p1 30030r8 [1, 0, 1, -105752574, -418583571128] 2 5971968  
30030.p2 30030r6 [1, 0, 1, -6887574, -5960607128] 4 2985984  
30030.p3 30030r5 [1, 0, 1, -2313909, 426827206] 6 1990656  
30030.p4 30030r3 [1, 0, 1, -1887574, 907392872] 2 1492992  
30030.p5 30030r2 [1, 0, 1, -1840059, 959624146] 12 995328  
30030.p6 30030r1 [1, 0, 1, -1839559, 960172346] 6 497664 \(\Gamma_0(N)\)-optimal
30030.p7 30030r4 [1, 0, 1, -1374209, 1457338286] 6 1990656  
30030.p8 30030r7 [1, 0, 1, 11977426, -32869643128] 2 5971968  

Rank

sage: E.rank()

The elliptic curves in class 30030.p have rank \(1\).

Modular form 30030.2.a.p

sage: E.q_eigenform(10)
\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + q^{13} - q^{14} - q^{15} + q^{16} - 6q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.