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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 4290bb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4290.bb7 | 4290bb1 | \([1, 0, 0, -11420, -450288]\) | \(164711681450297281/8097103872000\) | \(8097103872000\) | \([6]\) | \(13824\) | \(1.2362\) | \(\Gamma_0(N)\)-optimal |
4290.bb6 | 4290bb2 | \([1, 0, 0, -31900, 1610000]\) | \(3590017885052913601/954068544000000\) | \(954068544000000\) | \([2, 6]\) | \(27648\) | \(1.5828\) | |
4290.bb3 | 4290bb3 | \([1, 0, 0, -913820, -336308208]\) | \(84392862605474684114881/11228954880\) | \(11228954880\) | \([2]\) | \(41472\) | \(1.7855\) | |
4290.bb5 | 4290bb4 | \([1, 0, 0, -471900, 124722000]\) | \(11621808143080380273601/1335706803288000\) | \(1335706803288000\) | \([12]\) | \(55296\) | \(1.9294\) | |
4290.bb8 | 4290bb5 | \([1, 0, 0, 80420, 10438352]\) | \(57519563401957999679/80296734375000000\) | \(-80296734375000000\) | \([6]\) | \(55296\) | \(1.9294\) | |
4290.bb2 | 4290bb6 | \([1, 0, 0, -913900, -336246400]\) | \(84415028961834287121601/30783551683856400\) | \(30783551683856400\) | \([2, 2]\) | \(82944\) | \(2.1321\) | |
4290.bb1 | 4290bb7 | \([1, 0, 0, -1047000, -231922620]\) | \(126929854754212758768001/50235797102795981820\) | \(50235797102795981820\) | \([4]\) | \(165888\) | \(2.4787\) | |
4290.bb4 | 4290bb8 | \([1, 0, 0, -782080, -436614148]\) | \(-52902632853833942200321/51713453577420277500\) | \(-51713453577420277500\) | \([2]\) | \(165888\) | \(2.4787\) |
Rank
sage: E.rank()
The elliptic curves in class 4290bb have rank \(1\).
Complex multiplication
The elliptic curves in class 4290bb do not have complex multiplication.Modular form 4290.2.a.bb
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.