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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 6336.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6336.t1 | 6336bc3 | \([0, 0, 0, -5196, -129584]\) | \(649461896/72171\) | \(1724011610112\) | \([2]\) | \(8192\) | \(1.0811\) | |
6336.t2 | 6336bc2 | \([0, 0, 0, -1236, 14560]\) | \(69934528/9801\) | \(29265629184\) | \([2, 2]\) | \(4096\) | \(0.73453\) | |
6336.t3 | 6336bc1 | \([0, 0, 0, -1191, 15820]\) | \(4004529472/99\) | \(4618944\) | \([2]\) | \(2048\) | \(0.38795\) | \(\Gamma_0(N)\)-optimal |
6336.t4 | 6336bc4 | \([0, 0, 0, 2004, 78064]\) | \(37259704/131769\) | \(-3147681005568\) | \([2]\) | \(8192\) | \(1.0811\) |
Rank
sage: E.rank()
The elliptic curves in class 6336.t have rank \(1\).
Complex multiplication
The elliptic curves in class 6336.t do not have complex multiplication.Modular form 6336.2.a.t
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.