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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 31200p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31200.bh3 | 31200p1 | \([0, 1, 0, -406758, -56638512]\) | \(7442744143086784/2927948765625\) | \(2927948765625000000\) | \([2, 2]\) | \(589824\) | \(2.2419\) | \(\Gamma_0(N)\)-optimal |
| 31200.bh4 | 31200p2 | \([0, 1, 0, 1304367, -407419137]\) | \(3834800837445824/3342041015625\) | \(-213890625000000000000\) | \([2]\) | \(1179648\) | \(2.5885\) | |
| 31200.bh2 | 31200p3 | \([0, 1, 0, -2938008, 1897486488]\) | \(350584567631475848/8259273550125\) | \(66074188401000000000\) | \([2]\) | \(1179648\) | \(2.5885\) | |
| 31200.bh1 | 31200p4 | \([0, 1, 0, -5688008, -5221701012]\) | \(2543984126301795848/909361981125\) | \(7274895849000000000\) | \([2]\) | \(1179648\) | \(2.5885\) |
Rank
sage: E.rank()
The elliptic curves in class 31200p have rank \(0\).
Complex multiplication
The elliptic curves in class 31200p do not have complex multiplication.Modular form 31200.2.a.p
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.
