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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 10080bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10080.be3 | 10080bz1 | \([0, 0, 0, -178617, 29049176]\) | \(13507798771700416/3544416225\) | \(165368283393600\) | \([2, 2]\) | \(61440\) | \(1.7130\) | \(\Gamma_0(N)\)-optimal |
| 10080.be2 | 10080bz2 | \([0, 0, 0, -200667, 21424286]\) | \(2394165105226952/854262178245\) | \(318851649505589760\) | \([2]\) | \(122880\) | \(2.0596\) | |
| 10080.be1 | 10080bz3 | \([0, 0, 0, -2857692, 1859393216]\) | \(864335783029582144/59535\) | \(177770557440\) | \([4]\) | \(122880\) | \(2.0596\) | |
| 10080.be4 | 10080bz4 | \([0, 0, 0, -156747, 36428114]\) | \(-1141100604753992/875529151875\) | \(-326789504879040000\) | \([2]\) | \(122880\) | \(2.0596\) |
Rank
sage: E.rank()
The elliptic curves in class 10080bz have rank \(0\).
Complex multiplication
The elliptic curves in class 10080bz do not have complex multiplication.Modular form 10080.2.a.bz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
