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SageMath
E = EllipticCurve("da1")
E.isogeny_class()
Elliptic curves in class 210210da
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
210210.cs7 | 210210da1 | \([1, 1, 1, -559581, 153889203]\) | \(164711681450297281/8097103872000\) | \(952616173436928000\) | \([2]\) | \(3981312\) | \(2.2092\) | \(\Gamma_0(N)\)-optimal |
210210.cs6 | 210210da2 | \([1, 1, 1, -1563101, -553793101]\) | \(3590017885052913601/954068544000000\) | \(112245210133056000000\) | \([2, 2]\) | \(7962624\) | \(2.5557\) | |
210210.cs3 | 210210da3 | \([1, 1, 1, -44777181, 115308938163]\) | \(84392862605474684114881/11228954880\) | \(1321075312677120\) | \([2]\) | \(11943936\) | \(2.7585\) | |
210210.cs8 | 210210da4 | \([1, 1, 1, 3940579, -3576414157]\) | \(57519563401957999679/80296734375000000\) | \(-9446830502484375000000\) | \([2]\) | \(15925248\) | \(2.9023\) | |
210210.cs5 | 210210da5 | \([1, 1, 1, -23123101, -42802769101]\) | \(11621808143080380273601/1335706803288000\) | \(157144569700029912000\) | \([2]\) | \(15925248\) | \(2.9023\) | |
210210.cs2 | 210210da6 | \([1, 1, 1, -44781101, 115287734099]\) | \(84415028961834287121601/30783551683856400\) | \(3621654072054021603600\) | \([2, 2]\) | \(23887872\) | \(3.1050\) | |
210210.cs4 | 210210da7 | \([1, 1, 1, -38321921, 149720330843]\) | \(-52902632853833942200321/51713453577420277500\) | \(-6084036099929918227597500\) | \([2]\) | \(47775744\) | \(3.4516\) | |
210210.cs1 | 210210da8 | \([1, 1, 1, -51303001, 79498155659]\) | \(126929854754212758768001/50235797102795981820\) | \(5910191293346844465141180\) | \([2]\) | \(47775744\) | \(3.4516\) |
Rank
sage: E.rank()
The elliptic curves in class 210210da have rank \(1\).
Complex multiplication
The elliptic curves in class 210210da do not have complex multiplication.Modular form 210210.2.a.da
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.