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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 40362.bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40362.bb1 | 40362bc4 | \([1, 0, 0, -1291604, -565099500]\) | \(268498407453697/252\) | \(223650927612\) | \([2]\) | \(491520\) | \(1.9061\) | |
| 40362.bb2 | 40362bc6 | \([1, 0, 0, -878374, 313744610]\) | \(84448510979617/933897762\) | \(828837701452661922\) | \([2]\) | \(983040\) | \(2.2527\) | |
| 40362.bb3 | 40362bc3 | \([1, 0, 0, -99964, -4313716]\) | \(124475734657/63011844\) | \(55923243496597764\) | \([2, 2]\) | \(491520\) | \(1.9061\) | |
| 40362.bb4 | 40362bc2 | \([1, 0, 0, -80744, -8830416]\) | \(65597103937/63504\) | \(56360033758224\) | \([2, 2]\) | \(245760\) | \(1.5595\) | |
| 40362.bb5 | 40362bc1 | \([1, 0, 0, -3864, -204480]\) | \(-7189057/16128\) | \(-14313659367168\) | \([2]\) | \(122880\) | \(1.2129\) | \(\Gamma_0(N)\)-optimal |
| 40362.bb6 | 40362bc5 | \([1, 0, 0, 370926, -33226362]\) | \(6359387729183/4218578658\) | \(-3744004087563040098\) | \([2]\) | \(983040\) | \(2.2527\) |
Rank
sage: E.rank()
The elliptic curves in class 40362.bb have rank \(0\).
Complex multiplication
The elliptic curves in class 40362.bb do not have complex multiplication.Modular form 40362.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
