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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 174240cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 174240.dk3 | 174240cr1 | \([0, 0, 0, -6897, -74536]\) | \(438976/225\) | \(18597138753600\) | \([2, 2]\) | \(327680\) | \(1.2384\) | \(\Gamma_0(N)\)-optimal |
| 174240.dk2 | 174240cr2 | \([0, 0, 0, -61347, 5795174]\) | \(38614472/405\) | \(267798798051840\) | \([2]\) | \(655360\) | \(1.5850\) | |
| 174240.dk4 | 174240cr3 | \([0, 0, 0, 25773, -577654]\) | \(2863288/1875\) | \(-1239809250240000\) | \([2]\) | \(655360\) | \(1.5850\) | |
| 174240.dk1 | 174240cr4 | \([0, 0, 0, -88572, -10136896]\) | \(14526784/15\) | \(79347792015360\) | \([2]\) | \(655360\) | \(1.5850\) |
Rank
sage: E.rank()
The elliptic curves in class 174240cr have rank \(0\).
Complex multiplication
The elliptic curves in class 174240cr do not have complex multiplication.Modular form 174240.2.a.cr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
