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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 31680cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31680.eg3 | 31680cf1 | \([0, 0, 0, -100332, 2591056]\) | \(15781142246787/8722841600\) | \(61739295886540800\) | \([2]\) | \(331776\) | \(1.9123\) | \(\Gamma_0(N)\)-optimal |
| 31680.eg4 | 31680cf2 | \([0, 0, 0, 391188, 20482384]\) | \(935355271080573/566899520000\) | \(-4012451309813760000\) | \([2]\) | \(663552\) | \(2.2589\) | |
| 31680.eg1 | 31680cf3 | \([0, 0, 0, -6182892, 5917452624]\) | \(5066026756449723/11000000\) | \(56757583872000000\) | \([2]\) | \(995328\) | \(2.4616\) | |
| 31680.eg2 | 31680cf4 | \([0, 0, 0, -6113772, 6056217936]\) | \(-4898016158612283/236328125000\) | \(-1219401216000000000000\) | \([2]\) | \(1990656\) | \(2.8082\) |
Rank
sage: E.rank()
The elliptic curves in class 31680cf have rank \(0\).
Complex multiplication
The elliptic curves in class 31680cf do not have complex multiplication.Modular form 31680.2.a.cf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
