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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 199920dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 199920.k6 | 199920dd1 | \([0, -1, 0, -62736, 7452096]\) | \(-56667352321/16711680\) | \(-8053196555550720\) | \([2]\) | \(1179648\) | \(1.7665\) | \(\Gamma_0(N)\)-optimal |
| 199920.k5 | 199920dd2 | \([0, -1, 0, -1066256, 424113600]\) | \(278202094583041/16646400\) | \(8021738756505600\) | \([2, 2]\) | \(2359296\) | \(2.1131\) | |
| 199920.k2 | 199920dd3 | \([0, -1, 0, -17059856, 27127028160]\) | \(1139466686381936641/4080\) | \(1966112440320\) | \([2]\) | \(4718592\) | \(2.4597\) | |
| 199920.k4 | 199920dd4 | \([0, -1, 0, -1128976, 371478976]\) | \(330240275458561/67652010000\) | \(32600847665111040000\) | \([2, 2]\) | \(4718592\) | \(2.4597\) | |
| 199920.k7 | 199920dd5 | \([0, -1, 0, 2399024, 2225795776]\) | \(3168685387909439/6278181696900\) | \(-3025394886486376857600\) | \([2]\) | \(9437184\) | \(2.8063\) | |
| 199920.k3 | 199920dd6 | \([0, -1, 0, -5660496, -4852457280]\) | \(41623544884956481/2962701562500\) | \(1427697156614400000000\) | \([2, 2]\) | \(9437184\) | \(2.8063\) | |
| 199920.k8 | 199920dd7 | \([0, -1, 0, 5135184, -21184161984]\) | \(31077313442863199/420227050781250\) | \(-202503341250000000000000\) | \([2]\) | \(18874368\) | \(3.1528\) | |
| 199920.k1 | 199920dd8 | \([0, -1, 0, -88960496, -322925177280]\) | \(161572377633716256481/914742821250\) | \(440805696213980160000\) | \([2]\) | \(18874368\) | \(3.1528\) |
Rank
sage: E.rank()
The elliptic curves in class 199920dd have rank \(1\).
Complex multiplication
The elliptic curves in class 199920dd do not have complex multiplication.Modular form 199920.2.a.dd
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
