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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 333960.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 333960.p1 | 333960p4 | \([0, -1, 0, -127816, -17265620]\) | \(63649751618/1164375\) | \(4224535223040000\) | \([2]\) | \(1966080\) | \(1.7932\) | |
| 333960.p2 | 333960p2 | \([0, -1, 0, -16496, 411996]\) | \(273671716/119025\) | \(215920689177600\) | \([2, 2]\) | \(983040\) | \(1.4466\) | |
| 333960.p3 | 333960p1 | \([0, -1, 0, -14076, 647220]\) | \(680136784/345\) | \(156464267520\) | \([2]\) | \(491520\) | \(1.1000\) | \(\Gamma_0(N)\)-optimal |
| 333960.p4 | 333960p3 | \([0, -1, 0, 56104, 2996556]\) | \(5382838942/4197615\) | \(-15229605943326720\) | \([2]\) | \(1966080\) | \(1.7932\) |
Rank
sage: E.rank()
The elliptic curves in class 333960.p have rank \(2\).
Complex multiplication
The elliptic curves in class 333960.p do not have complex multiplication.Modular form 333960.2.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 LMFDB 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 LMFDB labels.
