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SageMath
E = EllipticCurve("gn1")
E.isogeny_class()
Elliptic curves in class 137280gn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 137280.e7 | 137280gn1 | \([0, -1, 0, -730881, -229816575]\) | \(164711681450297281/8097103872000\) | \(2122607197421568000\) | \([2]\) | \(2654208\) | \(2.2759\) | \(\Gamma_0(N)\)-optimal |
| 137280.e6 | 137280gn2 | \([0, -1, 0, -2041601, 826361601]\) | \(3590017885052913601/954068544000000\) | \(250103344398336000000\) | \([2, 2]\) | \(5308416\) | \(2.6225\) | |
| 137280.e3 | 137280gn3 | \([0, -1, 0, -58484481, -172131318015]\) | \(84392862605474684114881/11228954880\) | \(2943603148062720\) | \([2]\) | \(7962624\) | \(2.8252\) | |
| 137280.e5 | 137280gn4 | \([0, -1, 0, -30201601, 63887865601]\) | \(11621808143080380273601/1335706803288000\) | \(350147524241129472000\) | \([2]\) | \(10616832\) | \(2.9691\) | |
| 137280.e8 | 137280gn5 | \([0, -1, 0, 5146879, 5339289345]\) | \(57519563401957999679/80296734375000000\) | \(-21049307136000000000000\) | \([2]\) | \(10616832\) | \(2.9691\) | |
| 137280.e2 | 137280gn6 | \([0, -1, 0, -58489601, -172099667199]\) | \(84415028961834287121601/30783551683856400\) | \(8069723372612852121600\) | \([2, 2]\) | \(15925248\) | \(3.1718\) | |
| 137280.e1 | 137280gn7 | \([0, -1, 0, -67008001, -118677373439]\) | \(126929854754212758768001/50235797102795981820\) | \(13169012795715349858222080\) | \([2]\) | \(31850496\) | \(3.5184\) | |
| 137280.e4 | 137280gn8 | \([0, -1, 0, -50053121, -223496390655]\) | \(-52902632853833942200321/51713453577420277500\) | \(-13556371574599261224960000\) | \([2]\) | \(31850496\) | \(3.5184\) |
Rank
sage: E.rank()
The elliptic curves in class 137280gn have rank \(0\).
Complex multiplication
The elliptic curves in class 137280gn do not have complex multiplication.Modular form 137280.2.a.gn
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.
