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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 22386z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22386.x4 | 22386z1 | \([1, 0, 0, -3829, -239071]\) | \(-6208503067778257/21032186413056\) | \(-21032186413056\) | \([2]\) | \(64512\) | \(1.2426\) | \(\Gamma_0(N)\)-optimal |
| 22386.x3 | 22386z2 | \([1, 0, 0, -85749, -9659871]\) | \(69728644177980628177/100579396829184\) | \(100579396829184\) | \([2, 2]\) | \(129024\) | \(1.5892\) | |
| 22386.x2 | 22386z3 | \([1, 0, 0, -110709, -3584607]\) | \(150062694782364873937/81319429594096512\) | \(81319429594096512\) | \([2]\) | \(258048\) | \(1.9357\) | |
| 22386.x1 | 22386z4 | \([1, 0, 0, -1371509, -618338655]\) | \(285311789321435384726737/594905980032\) | \(594905980032\) | \([2]\) | \(258048\) | \(1.9357\) |
Rank
sage: E.rank()
The elliptic curves in class 22386z have rank \(1\).
Complex multiplication
The elliptic curves in class 22386z do not have complex multiplication.Modular form 22386.2.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.
