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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 6336p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6336.r3 | 6336p1 | \([0, 0, 0, -1191, -15820]\) | \(4004529472/99\) | \(4618944\) | \([2]\) | \(2048\) | \(0.38795\) | \(\Gamma_0(N)\)-optimal |
| 6336.r2 | 6336p2 | \([0, 0, 0, -1236, -14560]\) | \(69934528/9801\) | \(29265629184\) | \([2, 2]\) | \(4096\) | \(0.73453\) | |
| 6336.r1 | 6336p3 | \([0, 0, 0, -5196, 129584]\) | \(649461896/72171\) | \(1724011610112\) | \([2]\) | \(8192\) | \(1.0811\) | |
| 6336.r4 | 6336p4 | \([0, 0, 0, 2004, -78064]\) | \(37259704/131769\) | \(-3147681005568\) | \([2]\) | \(8192\) | \(1.0811\) |
Rank
sage: E.rank()
The elliptic curves in class 6336p have rank \(0\).
Complex multiplication
The elliptic curves in class 6336p do not have complex multiplication.Modular form 6336.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
