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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 12635c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12635.c3 | 12635c1 | \([1, -1, 1, -5122, 142104]\) | \(315821241/665\) | \(31285510865\) | \([4]\) | \(11520\) | \(0.89891\) | \(\Gamma_0(N)\)-optimal |
12635.c2 | 12635c2 | \([1, -1, 1, -6927, 34526]\) | \(781229961/442225\) | \(20804864725225\) | \([2, 2]\) | \(23040\) | \(1.2455\) | |
12635.c1 | 12635c3 | \([1, -1, 1, -70102, -7091614]\) | \(809818183161/4561235\) | \(214587319023035\) | \([2]\) | \(46080\) | \(1.5921\) | |
12635.c4 | 12635c4 | \([1, -1, 1, 27368, 254014]\) | \(48188806119/28511875\) | \(-1341366278336875\) | \([2]\) | \(46080\) | \(1.5921\) |
Rank
sage: E.rank()
The elliptic curves in class 12635c have rank \(1\).
Complex multiplication
The elliptic curves in class 12635c do not have complex multiplication.Modular form 12635.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.