Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("30030.bt1")
sage: E.isogeny_class()

Elliptic curves in class 30030.bt

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
30030.bt1 30030bt8 [1, 0, 0, -195003991, 1045763011475] 2 9289728  
30030.bt2 30030bt5 [1, 0, 0, -194882701, 1047131714681] 6 3096576  
30030.bt3 30030bt7 [1, 0, 0, -177737371, -908394585049] 2 9289728  
30030.bt4 30030bt6 [1, 0, 0, -16956121, 2366883701] 4 4644864  
30030.bt5 30030bt4 [1, 0, 0, -12369181, 15826636745] 6 3096576  
30030.bt6 30030bt2 [1, 0, 0, -12180181, 16360637345] 12 1548288  
30030.bt7 30030bt1 [1, 0, 0, -749461, 263897441] 12 774144 \(\Gamma_0(N)\)-optimal
30030.bt8 30030bt3 [1, 0, 0, 4220699, 295790705] 4 2322432  

Rank

sage: E.rank()

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

Modular form 30030.2.a.bt

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.