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SageMath
E = EllipticCurve("pa1")
E.isogeny_class()
Elliptic curves in class 388080.pa
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.pa1 | 388080pa4 | \([0, 0, 0, -72441747, -237318440814]\) | \(119678115308998401/1925\) | \(676248710860800\) | \([2]\) | \(25165824\) | \(2.8443\) | |
| 388080.pa2 | 388080pa3 | \([0, 0, 0, -4915827, -3034700046]\) | \(37397086385121/10316796875\) | \(3624270434769600000000\) | \([2]\) | \(25165824\) | \(2.8443\) | |
| 388080.pa3 | 388080pa2 | \([0, 0, 0, -4527747, -3707863614]\) | \(29220958012401/3705625\) | \(1301778768407040000\) | \([2, 2]\) | \(12582912\) | \(2.4977\) | |
| 388080.pa4 | 388080pa1 | \([0, 0, 0, -258867, -68216526]\) | \(-5461074081/2562175\) | \(-900087034155724800\) | \([2]\) | \(6291456\) | \(2.1512\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 388080.pa have rank \(0\).
Complex multiplication
The elliptic curves in class 388080.pa do not have complex multiplication.Modular form 388080.2.a.pa
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.
