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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 53067q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53067.r6 | 53067q1 | \([1, 0, 1, 17320, -191887]\) | \(103823/63\) | \(-348698753787447\) | \([2]\) | \(165888\) | \(1.4797\) | \(\Gamma_0(N)\)-optimal |
| 53067.r5 | 53067q2 | \([1, 0, 1, -71125, -1571629]\) | \(7189057/3969\) | \(21968021488609161\) | \([2, 2]\) | \(331776\) | \(1.8262\) | |
| 53067.r3 | 53067q3 | \([1, 0, 1, -690240, 219328603]\) | \(6570725617/45927\) | \(254201391511048863\) | \([2]\) | \(663552\) | \(2.1728\) | |
| 53067.r2 | 53067q4 | \([1, 0, 1, -867130, -310421569]\) | \(13027640977/21609\) | \(119603672549094321\) | \([2, 2]\) | \(663552\) | \(2.1728\) | |
| 53067.r4 | 53067q5 | \([1, 0, 1, -601795, -503903851]\) | \(-4354703137/17294403\) | \(-95722805930125154907\) | \([2]\) | \(1327104\) | \(2.5194\) | |
| 53067.r1 | 53067q6 | \([1, 0, 1, -13868545, -19880151427]\) | \(53297461115137/147\) | \(813630425504043\) | \([2]\) | \(1327104\) | \(2.5194\) |
Rank
sage: E.rank()
The elliptic curves in class 53067q have rank \(0\).
Complex multiplication
The elliptic curves in class 53067q do not have complex multiplication.Modular form 53067.2.a.q
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
