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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 88200.dz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 88200.dz1 | 88200bw4 | \([0, 0, 0, -41193075, -101761497250]\) | \(5633270409316/14175\) | \(19451756242800000000\) | \([2]\) | \(4718592\) | \(2.9395\) | |
| 88200.dz2 | 88200bw3 | \([0, 0, 0, -7236075, 5483969750]\) | \(30534944836/8203125\) | \(11256803381250000000000\) | \([2]\) | \(4718592\) | \(2.9395\) | |
| 88200.dz3 | 88200bw2 | \([0, 0, 0, -2605575, -1549759750]\) | \(5702413264/275625\) | \(94557148402500000000\) | \([2, 2]\) | \(2359296\) | \(2.5929\) | |
| 88200.dz4 | 88200bw1 | \([0, 0, 0, 95550, -93853375]\) | \(4499456/180075\) | \(-3861083559768750000\) | \([2]\) | \(1179648\) | \(2.2463\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 88200.dz have rank \(1\).
Complex multiplication
The elliptic curves in class 88200.dz do not have complex multiplication.Modular form 88200.2.a.dz
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
