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SageMath
sage: E = EllipticCurve("km1")
sage: E.isogeny_class()
Elliptic curves in class 388080km
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 388080.km6 | 388080km1 | [0, 0, 0, 246813, 13709786146] | [2] | 23592960 | \(\Gamma_0(N)\)-optimal |
| 388080.km5 | 388080km2 | [0, 0, 0, -84460467, 293464049074] | [2, 2] | 47185920 | |
| 388080.km2 | 388080km3 | [0, 0, 0, -1344697347, 18979500363586] | [2, 2] | 94371840 | |
| 388080.km4 | 388080km4 | [0, 0, 0, -179540067, -488299438046] | [2] | 94371840 | |
| 388080.km1 | 388080km5 | [0, 0, 0, -21515155347, 1214688284695186] | [4] | 188743680 | |
| 388080.km3 | 388080km6 | [0, 0, 0, -1338029427, 19177040160754] | [2] | 188743680 |
Rank
sage: E.rank()
The elliptic curves in class 388080km have rank \(1\).
Complex multiplication
The elliptic curves in class 388080km do not have complex multiplication.Modular form 388080.2.a.km
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
