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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 112.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
112.c1 | 112c6 | \([0, -1, 0, -43688, 3529328]\) | \(2251439055699625/25088\) | \(102760448\) | \([2]\) | \(144\) | \(1.1062\) | |
112.c2 | 112c5 | \([0, -1, 0, -2728, 55920]\) | \(-548347731625/1835008\) | \(-7516192768\) | \([2]\) | \(72\) | \(0.75968\) | |
112.c3 | 112c4 | \([0, -1, 0, -568, 4464]\) | \(4956477625/941192\) | \(3855122432\) | \([2]\) | \(48\) | \(0.55694\) | |
112.c4 | 112c2 | \([0, -1, 0, -168, -784]\) | \(128787625/98\) | \(401408\) | \([2]\) | \(16\) | \(0.0076359\) | |
112.c5 | 112c1 | \([0, -1, 0, -8, -16]\) | \(-15625/28\) | \(-114688\) | \([2]\) | \(8\) | \(-0.33894\) | \(\Gamma_0(N)\)-optimal |
112.c6 | 112c3 | \([0, -1, 0, 72, 368]\) | \(9938375/21952\) | \(-89915392\) | \([2]\) | \(24\) | \(0.21037\) |
Rank
sage: E.rank()
The elliptic curves in class 112.c have rank \(0\).
Complex multiplication
The elliptic curves in class 112.c do not have complex multiplication.Modular form 112.2.a.c
sage: E.q_eigenform(10)
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.