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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 8400.cn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8400.cn1 | 8400ce7 | \([0, 1, 0, -2580408, -1004608812]\) | \(29689921233686449/10380965400750\) | \(664381785648000000000\) | \([2]\) | \(331776\) | \(2.6969\) | |
8400.cn2 | 8400ce4 | \([0, 1, 0, -2304408, -1347208812]\) | \(21145699168383889/2593080\) | \(165957120000000\) | \([2]\) | \(110592\) | \(2.1476\) | |
8400.cn3 | 8400ce6 | \([0, 1, 0, -1080408, 420391188]\) | \(2179252305146449/66177562500\) | \(4235364000000000000\) | \([2, 2]\) | \(165888\) | \(2.3503\) | |
8400.cn4 | 8400ce3 | \([0, 1, 0, -1072408, 427095188]\) | \(2131200347946769/2058000\) | \(131712000000000\) | \([2]\) | \(82944\) | \(2.0037\) | |
8400.cn5 | 8400ce2 | \([0, 1, 0, -144408, -20968812]\) | \(5203798902289/57153600\) | \(3657830400000000\) | \([2, 2]\) | \(55296\) | \(1.8010\) | |
8400.cn6 | 8400ce5 | \([0, 1, 0, -32408, -52552812]\) | \(-58818484369/18600435000\) | \(-1190427840000000000\) | \([2]\) | \(110592\) | \(2.1476\) | |
8400.cn7 | 8400ce1 | \([0, 1, 0, -16408, 279188]\) | \(7633736209/3870720\) | \(247726080000000\) | \([2]\) | \(27648\) | \(1.4544\) | \(\Gamma_0(N)\)-optimal |
8400.cn8 | 8400ce8 | \([0, 1, 0, 291592, 1416463188]\) | \(42841933504271/13565917968750\) | \(-868218750000000000000\) | \([2]\) | \(331776\) | \(2.6969\) |
Rank
sage: E.rank()
The elliptic curves in class 8400.cn have rank \(0\).
Complex multiplication
The elliptic curves in class 8400.cn do not have complex multiplication.Modular form 8400.2.a.cn
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.