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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 5355d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5355.n5 | 5355d1 | \([1, -1, 0, 315, -5400]\) | \(4733169839/19518975\) | \(-14229332775\) | \([2]\) | \(4096\) | \(0.63084\) | \(\Gamma_0(N)\)-optimal |
| 5355.n4 | 5355d2 | \([1, -1, 0, -3330, -64449]\) | \(5602762882081/716900625\) | \(522620555625\) | \([2, 2]\) | \(8192\) | \(0.97741\) | |
| 5355.n2 | 5355d3 | \([1, -1, 0, -51525, -4488750]\) | \(20751759537944401/418359375\) | \(304983984375\) | \([2]\) | \(16384\) | \(1.3240\) | |
| 5355.n3 | 5355d4 | \([1, -1, 0, -13455, 536976]\) | \(369543396484081/45120132225\) | \(32892576392025\) | \([2, 2]\) | \(16384\) | \(1.3240\) | |
| 5355.n1 | 5355d5 | \([1, -1, 0, -208530, 36703881]\) | \(1375634265228629281/24990412335\) | \(18218010592215\) | \([2]\) | \(32768\) | \(1.6706\) | |
| 5355.n6 | 5355d6 | \([1, -1, 0, 19620, 2739771]\) | \(1145725929069119/5127181719135\) | \(-3737715473249415\) | \([2]\) | \(32768\) | \(1.6706\) |
Rank
sage: E.rank()
The elliptic curves in class 5355d have rank \(0\).
Complex multiplication
The elliptic curves in class 5355d do not have complex multiplication.Modular form 5355.2.a.d
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.
