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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 5040.u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.u1 | 5040p4 | \([0, 0, 0, -4107, 101194]\) | \(10262905636/13125\) | \(9797760000\) | \([4]\) | \(4096\) | \(0.82502\) | |
5040.u2 | 5040p3 | \([0, 0, 0, -3027, -63614]\) | \(4108974916/36015\) | \(26885053440\) | \([2]\) | \(4096\) | \(0.82502\) | |
5040.u3 | 5040p2 | \([0, 0, 0, -327, 646]\) | \(20720464/11025\) | \(2057529600\) | \([2, 2]\) | \(2048\) | \(0.47845\) | |
5040.u4 | 5040p1 | \([0, 0, 0, 78, 79]\) | \(4499456/2835\) | \(-33067440\) | \([2]\) | \(1024\) | \(0.13187\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5040.u have rank \(1\).
Complex multiplication
The elliptic curves in class 5040.u do not have complex multiplication.Modular form 5040.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.