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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 12870p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.c7 | 12870p1 | \([1, -1, 0, -102780, 12157776]\) | \(164711681450297281/8097103872000\) | \(5902788722688000\) | \([2]\) | \(110592\) | \(1.7855\) | \(\Gamma_0(N)\)-optimal |
| 12870.c6 | 12870p2 | \([1, -1, 0, -287100, -43470000]\) | \(3590017885052913601/954068544000000\) | \(695515968576000000\) | \([2, 2]\) | \(221184\) | \(2.1321\) | |
| 12870.c3 | 12870p3 | \([1, -1, 0, -8224380, 9080321616]\) | \(84392862605474684114881/11228954880\) | \(8185908107520\) | \([6]\) | \(331776\) | \(2.3348\) | |
| 12870.c5 | 12870p4 | \([1, -1, 0, -4247100, -3367494000]\) | \(11621808143080380273601/1335706803288000\) | \(973730259596952000\) | \([2]\) | \(442368\) | \(2.4787\) | |
| 12870.c8 | 12870p5 | \([1, -1, 0, 723780, -281835504]\) | \(57519563401957999679/80296734375000000\) | \(-58536319359375000000\) | \([2]\) | \(442368\) | \(2.4787\) | |
| 12870.c2 | 12870p6 | \([1, -1, 0, -8225100, 9078652800]\) | \(84415028961834287121601/30783551683856400\) | \(22441209177531315600\) | \([2, 6]\) | \(663552\) | \(2.6814\) | |
| 12870.c1 | 12870p7 | \([1, -1, 0, -9423000, 6261910740]\) | \(126929854754212758768001/50235797102795981820\) | \(36621896087938270746780\) | \([6]\) | \(1327104\) | \(3.0280\) | |
| 12870.c4 | 12870p8 | \([1, -1, 0, -7038720, 11788581996]\) | \(-52902632853833942200321/51713453577420277500\) | \(-37699107657939382297500\) | \([6]\) | \(1327104\) | \(3.0280\) |
Rank
sage: E.rank()
The elliptic curves in class 12870p have rank \(0\).
Complex multiplication
The elliptic curves in class 12870p do not have complex multiplication.Modular form 12870.2.a.p
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.
