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SageMath
E = EllipticCurve("el1")
E.isogeny_class()
Elliptic curves in class 64350el
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 64350.fe7 | 64350el1 | \([1, -1, 1, -2569505, 1517152497]\) | \(164711681450297281/8097103872000\) | \(92231073792000000000\) | \([2]\) | \(2654208\) | \(2.5902\) | \(\Gamma_0(N)\)-optimal |
| 64350.fe6 | 64350el2 | \([1, -1, 1, -7177505, -5440927503]\) | \(3590017885052913601/954068544000000\) | \(10867437009000000000000\) | \([2, 2]\) | \(5308416\) | \(2.9368\) | |
| 64350.fe3 | 64350el3 | \([1, -1, 1, -205609505, 1134834592497]\) | \(84392862605474684114881/11228954880\) | \(127904814180000000\) | \([2]\) | \(7962624\) | \(3.1395\) | |
| 64350.fe8 | 64350el4 | \([1, -1, 1, 18094495, -35211343503]\) | \(57519563401957999679/80296734375000000\) | \(-914629989990234375000000\) | \([2]\) | \(10616832\) | \(3.2834\) | |
| 64350.fe5 | 64350el5 | \([1, -1, 1, -106177505, -421042927503]\) | \(11621808143080380273601/1335706803288000\) | \(15214535306202375000000\) | \([2]\) | \(10616832\) | \(3.2834\) | |
| 64350.fe2 | 64350el6 | \([1, -1, 1, -205627505, 1134625972497]\) | \(84415028961834287121601/30783551683856400\) | \(350643893398926806250000\) | \([2, 2]\) | \(15925248\) | \(3.4861\) | |
| 64350.fe4 | 64350el7 | \([1, -1, 1, -175968005, 1473396781497]\) | \(-52902632853833942200321/51713453577420277500\) | \(-589048557155302848398437500\) | \([2]\) | \(31850496\) | \(3.8327\) | |
| 64350.fe1 | 64350el8 | \([1, -1, 1, -235575005, 782503267497]\) | \(126929854754212758768001/50235797102795981820\) | \(572217126374035480418437500\) | \([2]\) | \(31850496\) | \(3.8327\) |
Rank
sage: E.rank()
The elliptic curves in class 64350el have rank \(0\).
Complex multiplication
The elliptic curves in class 64350el do not have complex multiplication.Modular form 64350.2.a.el
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.
