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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 333270.n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 333270.n1 | 333270n7 | \([1, -1, 0, -30713310, 41225237550]\) | \(29689921233686449/10380965400750\) | \(1120294717056357019710750\) | \([2]\) | \(58392576\) | \(3.3161\) | |
| 333270.n2 | 333270n4 | \([1, -1, 0, -27428220, 55296592056]\) | \(21145699168383889/2593080\) | \(279840430322079480\) | \([2]\) | \(19464192\) | \(2.7668\) | |
| 333270.n3 | 333270n6 | \([1, -1, 0, -12859560, -17274359700]\) | \(2179252305146449/66177562500\) | \(7141760982178070062500\) | \([2, 2]\) | \(29196288\) | \(2.9695\) | |
| 333270.n4 | 333270n3 | \([1, -1, 0, -12764340, -17549564544]\) | \(2131200347946769/2058000\) | \(222095579620698000\) | \([2]\) | \(14598144\) | \(2.6229\) | |
| 333270.n5 | 333270n2 | \([1, -1, 0, -1718820, 859508496]\) | \(5203798902289/57153600\) | \(6167911525466241600\) | \([2, 2]\) | \(9732096\) | \(2.4202\) | |
| 333270.n6 | 333270n5 | \([1, -1, 0, -385740, 2157661800]\) | \(-58818484369/18600435000\) | \(-2007324777707540235000\) | \([2]\) | \(19464192\) | \(2.7668\) | |
| 333270.n7 | 333270n1 | \([1, -1, 0, -195300, -11640240]\) | \(7633736209/3870720\) | \(417720992200888320\) | \([2]\) | \(4866048\) | \(2.0736\) | \(\Gamma_0(N)\)-optimal |
| 333270.n8 | 333270n8 | \([1, -1, 0, 3470670, -58161989574]\) | \(42841933504271/13565917968750\) | \(-1464008947695030761718750\) | \([2]\) | \(58392576\) | \(3.3161\) |
Rank
sage: E.rank()
The elliptic curves in class 333270.n have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.n do not have complex multiplication.Modular form 333270.2.a.n
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
