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SageMath
E = EllipticCurve("ii1")
E.isogeny_class()
Elliptic curves in class 327600ii
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 327600.ii3 | 327600ii1 | \([0, 0, 0, -25275, -1385750]\) | \(38272753/4368\) | \(203793408000000\) | \([2]\) | \(1179648\) | \(1.4780\) | \(\Gamma_0(N)\)-optimal |
| 327600.ii2 | 327600ii2 | \([0, 0, 0, -97275, 10206250]\) | \(2181825073/298116\) | \(13908900096000000\) | \([2, 2]\) | \(2359296\) | \(1.8246\) | |
| 327600.ii1 | 327600ii3 | \([0, 0, 0, -1501275, 707994250]\) | \(8020417344913/187278\) | \(8737642368000000\) | \([2]\) | \(4718592\) | \(2.1711\) | |
| 327600.ii4 | 327600ii4 | \([0, 0, 0, 154725, 54306250]\) | \(8780064047/32388174\) | \(-1511102646144000000\) | \([2]\) | \(4718592\) | \(2.1711\) |
Rank
sage: E.rank()
The elliptic curves in class 327600ii have rank \(1\).
Complex multiplication
The elliptic curves in class 327600ii do not have complex multiplication.Modular form 327600.2.a.ii
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.
