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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 43350dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 43350.cv8 | 43350dd1 | \([1, 0, 0, 10687, 1306617]\) | \(357911/2160\) | \(-814642953750000\) | \([2]\) | \(221184\) | \(1.5430\) | \(\Gamma_0(N)\)-optimal |
| 43350.cv6 | 43350dd2 | \([1, 0, 0, -133813, 17057117]\) | \(702595369/72900\) | \(27494199689062500\) | \([2, 2]\) | \(442368\) | \(1.8896\) | |
| 43350.cv7 | 43350dd3 | \([1, 0, 0, -97688, -38467008]\) | \(-273359449/1536000\) | \(-579301656000000000\) | \([2]\) | \(663552\) | \(2.0923\) | |
| 43350.cv5 | 43350dd4 | \([1, 0, 0, -495063, -115521633]\) | \(35578826569/5314410\) | \(2004327157332656250\) | \([2]\) | \(884736\) | \(2.2362\) | |
| 43350.cv4 | 43350dd5 | \([1, 0, 0, -2084563, 1158245867]\) | \(2656166199049/33750\) | \(12728796152343750\) | \([2]\) | \(884736\) | \(2.2362\) | |
| 43350.cv3 | 43350dd6 | \([1, 0, 0, -2409688, -1437227008]\) | \(4102915888729/9000000\) | \(3394345640625000000\) | \([2, 2]\) | \(1327104\) | \(2.4389\) | |
| 43350.cv1 | 43350dd7 | \([1, 0, 0, -38534688, -92074852008]\) | \(16778985534208729/81000\) | \(30549110765625000\) | \([2]\) | \(2654208\) | \(2.7855\) | |
| 43350.cv2 | 43350dd8 | \([1, 0, 0, -3276688, -310994008]\) | \(10316097499609/5859375000\) | \(2209860443115234375000\) | \([2]\) | \(2654208\) | \(2.7855\) |
Rank
sage: E.rank()
The elliptic curves in class 43350dd have rank \(1\).
Complex multiplication
The elliptic curves in class 43350dd do not have complex multiplication.Modular form 43350.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
