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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 33930j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33930.g4 | 33930j1 | \([1, -1, 0, -93015, -10893155]\) | \(122083727651299441/32242728960\) | \(23504949411840\) | \([2]\) | \(163840\) | \(1.5501\) | \(\Gamma_0(N)\)-optimal |
| 33930.g3 | 33930j2 | \([1, -1, 0, -104535, -8015459]\) | \(173294065906331761/61964605497600\) | \(45172197407750400\) | \([2, 2]\) | \(327680\) | \(1.8967\) | |
| 33930.g6 | 33930j3 | \([1, -1, 0, 316665, -56621939]\) | \(4817210305461175439/4682306425314960\) | \(-3413401384054605840\) | \([2]\) | \(655360\) | \(2.2432\) | |
| 33930.g2 | 33930j4 | \([1, -1, 0, -710055, 224625325]\) | \(54309086480107021681/1575939143610000\) | \(1148859635691690000\) | \([2, 2]\) | \(655360\) | \(2.2432\) | |
| 33930.g5 | 33930j5 | \([1, -1, 0, 172125, 745287961]\) | \(773618103830753999/329643718157812500\) | \(-240310270537045312500\) | \([2]\) | \(1310720\) | \(2.5898\) | |
| 33930.g1 | 33930j6 | \([1, -1, 0, -11280555, 14585706625]\) | \(217764763259392950709681/191615146362900\) | \(139687441698554100\) | \([2]\) | \(1310720\) | \(2.5898\) |
Rank
sage: E.rank()
The elliptic curves in class 33930j have rank \(1\).
Complex multiplication
The elliptic curves in class 33930j do not have complex multiplication.Modular form 33930.2.a.j
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.
