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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 3744p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3744.o3 | 3744p1 | \([0, 0, 0, -2109, -37100]\) | \(22235451328/123201\) | \(5748065856\) | \([2, 2]\) | \(3072\) | \(0.71545\) | \(\Gamma_0(N)\)-optimal |
| 3744.o1 | 3744p2 | \([0, 0, 0, -33699, -2381078]\) | \(11339065490696/351\) | \(131010048\) | \([2]\) | \(6144\) | \(1.0620\) | |
| 3744.o2 | 3744p3 | \([0, 0, 0, -3324, 10528]\) | \(1360251712/771147\) | \(2302632603648\) | \([4]\) | \(6144\) | \(1.0620\) | |
| 3744.o4 | 3744p4 | \([0, 0, 0, -939, -78050]\) | \(-245314376/6908733\) | \(-2578670774784\) | \([2]\) | \(6144\) | \(1.0620\) |
Rank
sage: E.rank()
The elliptic curves in class 3744p have rank \(0\).
Complex multiplication
The elliptic curves in class 3744p do not have complex multiplication.Modular form 3744.2.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.
