Properties

Label 112c
Number of curves 6
Conductor 112
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("112.c1")
sage: E.isogeny_class()

Elliptic curves in class 112c

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
112.c5 112c1 [0, -1, 0, -8, -16] 2 8 \(\Gamma_0(N)\)-optimal
112.c4 112c2 [0, -1, 0, -168, -784] 2 16  
112.c6 112c3 [0, -1, 0, 72, 368] 2 24  
112.c3 112c4 [0, -1, 0, -568, 4464] 2 48  
112.c2 112c5 [0, -1, 0, -2728, 55920] 2 72  
112.c1 112c6 [0, -1, 0, -43688, 3529328] 2 144  

Rank

sage: E.rank()

The elliptic curves in class 112c have rank \(0\).

Modular form 112.2.a.c

sage: E.q_eigenform(10)
\( q + 2q^{3} - q^{7} + q^{9} - 4q^{13} + 6q^{17} - 2q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.