Properties

Label 1575g
Number of curves 6
Conductor 1575
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("1575.c1")
sage: E.isogeny_class()

Elliptic curves in class 1575g

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
1575.c6 1575g1 [1, -1, 1, 220, 222] 2 512 \(\Gamma_0(N)\)-optimal
1575.c5 1575g2 [1, -1, 1, -905, 2472] 4 1024  
1575.c3 1575g3 [1, -1, 1, -8780, -312528] 2 2048  
1575.c2 1575g4 [1, -1, 1, -11030, 447972] 4 2048  
1575.c1 1575g5 [1, -1, 1, -176405, 28561722] 2 4096  
1575.c4 1575g6 [1, -1, 1, -7655, 724722] 2 4096  

Rank

sage: E.rank()

The elliptic curves in class 1575g have rank \(1\).

Modular form 1575.2.a.c

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

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.