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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 5040.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5040.f1 | 5040u3 | \([0, 0, 0, -6723, 185922]\) | \(416832723/56000\) | \(4514807808000\) | \([2]\) | \(6912\) | \(1.1557\) | |
| 5040.f2 | 5040u1 | \([0, 0, 0, -1683, -26542]\) | \(4767078987/6860\) | \(758661120\) | \([2]\) | \(2304\) | \(0.60635\) | \(\Gamma_0(N)\)-optimal |
| 5040.f3 | 5040u2 | \([0, 0, 0, -1203, -41998]\) | \(-1740992427/5882450\) | \(-650551910400\) | \([2]\) | \(4608\) | \(0.95293\) | |
| 5040.f4 | 5040u4 | \([0, 0, 0, 10557, 984258]\) | \(1613964717/6125000\) | \(-493807104000000\) | \([2]\) | \(13824\) | \(1.5022\) |
Rank
sage: E.rank()
The elliptic curves in class 5040.f have rank \(0\).
Complex multiplication
The elliptic curves in class 5040.f do not have complex multiplication.Modular form 5040.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
