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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 73920.ek
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 73920.ek1 | 73920gk4 | \([0, 1, 0, -36801121, -85941253345]\) | \(21026497979043461623321/161783881875\) | \(42410673930240000\) | \([2]\) | \(3932160\) | \(2.7830\) | |
| 73920.ek2 | 73920gk2 | \([0, 1, 0, -2301601, -1341530401]\) | \(5143681768032498601/14238434358225\) | \(3732520136402534400\) | \([2, 2]\) | \(1966080\) | \(2.4364\) | |
| 73920.ek3 | 73920gk3 | \([0, 1, 0, -1394401, -2408941921]\) | \(-1143792273008057401/8897444448004035\) | \(-2332411677377569751040\) | \([2]\) | \(3932160\) | \(2.7830\) | |
| 73920.ek4 | 73920gk1 | \([0, 1, 0, -202081, -2456545]\) | \(3481467828171481/2005331497785\) | \(525685620155351040\) | \([2]\) | \(983040\) | \(2.0899\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 73920.ek have rank \(1\).
Complex multiplication
The elliptic curves in class 73920.ek do not have complex multiplication.Modular form 73920.2.a.ek
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.
