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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 21450.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21450.r1 | 21450d8 | \([1, 1, 0, -26175000, -28990327500]\) | \(126929854754212758768001/50235797102795981820\) | \(784934329731187215937500\) | \([2]\) | \(3981312\) | \(3.2834\) | |
| 21450.r2 | 21450d6 | \([1, 1, 0, -22847500, -42030800000]\) | \(84415028961834287121601/30783551683856400\) | \(480992995060256250000\) | \([2, 2]\) | \(1990656\) | \(2.9368\) | |
| 21450.r3 | 21450d3 | \([1, 1, 0, -22845500, -42038526000]\) | \(84392862605474684114881/11228954880\) | \(175452420000000\) | \([2]\) | \(995328\) | \(2.5902\) | |
| 21450.r4 | 21450d7 | \([1, 1, 0, -19552000, -54576768500]\) | \(-52902632853833942200321/51713453577420277500\) | \(-808022712147191835937500\) | \([2]\) | \(3981312\) | \(3.2834\) | |
| 21450.r5 | 21450d5 | \([1, 1, 0, -11797500, 15590250000]\) | \(11621808143080380273601/1335706803288000\) | \(20870418801375000000\) | \([2]\) | \(1327104\) | \(2.7341\) | |
| 21450.r6 | 21450d2 | \([1, 1, 0, -797500, 201250000]\) | \(3590017885052913601/954068544000000\) | \(14907321000000000000\) | \([2, 2]\) | \(663552\) | \(2.3875\) | |
| 21450.r7 | 21450d1 | \([1, 1, 0, -285500, -56286000]\) | \(164711681450297281/8097103872000\) | \(126517248000000000\) | \([2]\) | \(331776\) | \(2.0409\) | \(\Gamma_0(N)\)-optimal |
| 21450.r8 | 21450d4 | \([1, 1, 0, 2010500, 1304794000]\) | \(57519563401957999679/80296734375000000\) | \(-1254636474609375000000\) | \([2]\) | \(1327104\) | \(2.7341\) |
Rank
sage: E.rank()
The elliptic curves in class 21450.r have rank \(1\).
Complex multiplication
The elliptic curves in class 21450.r do not have complex multiplication.Modular form 21450.2.a.r
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
