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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 55470z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55470.ba4 | 55470z1 | \([1, 1, 1, 1232320, -681394975]\) | \(32740359775271/50724864000\) | \(-320650280955150336000\) | \([4]\) | \(3548160\) | \(2.6200\) | \(\Gamma_0(N)\)-optimal |
| 55470.ba3 | 55470z2 | \([1, 1, 1, -8234560, -6906815263]\) | \(9768641617435609/2396304000000\) | \(15147907559770896000000\) | \([2, 2]\) | \(7096320\) | \(2.9666\) | |
| 55470.ba2 | 55470z3 | \([1, 1, 1, -45214560, 111251680737]\) | \(1617141066657115609/89723013444000\) | \(567171741829831830756000\) | \([2]\) | \(14192640\) | \(3.3132\) | |
| 55470.ba1 | 55470z4 | \([1, 1, 1, -122724640, -523302872095]\) | \(32337636827233520089/3023437500000\) | \(19112246093460937500000\) | \([2]\) | \(14192640\) | \(3.3132\) |
Rank
sage: E.rank()
The elliptic curves in class 55470z have rank \(0\).
Complex multiplication
The elliptic curves in class 55470z do not have complex multiplication.Modular form 55470.2.a.z
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 Cremona 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 Cremona labels.
