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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 174240r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 174240.g3 | 174240r1 | \([0, 0, 0, -33033, -2225432]\) | \(48228544/2025\) | \(167374248782400\) | \([2, 2]\) | \(737280\) | \(1.4940\) | \(\Gamma_0(N)\)-optimal |
| 174240.g2 | 174240r2 | \([0, 0, 0, -87483, 6998398]\) | \(111980168/32805\) | \(21691702642199040\) | \([2]\) | \(1474560\) | \(1.8406\) | |
| 174240.g4 | 174240r3 | \([0, 0, 0, 15972, -8262848]\) | \(85184/5625\) | \(-29755422005760000\) | \([2]\) | \(1474560\) | \(1.8406\) | |
| 174240.g1 | 174240r4 | \([0, 0, 0, -523083, -145614062]\) | \(23937672968/45\) | \(29755422005760\) | \([2]\) | \(1474560\) | \(1.8406\) |
Rank
sage: E.rank()
The elliptic curves in class 174240r have rank \(0\).
Complex multiplication
The elliptic curves in class 174240r do not have complex multiplication.Modular form 174240.2.a.r
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.
