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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 169050bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169050.jc4 | 169050bz1 | \([1, 0, 0, -4176217963, 90569481434417]\) | \(4381924769947287308715481/608122186185572352000\) | \(1117890110664787525632000000000\) | \([2]\) | \(371589120\) | \(4.4923\) | \(\Gamma_0(N)\)-optimal |
| 169050.jc2 | 169050bz2 | \([1, 0, 0, -64412505963, 6292076039898417]\) | \(16077778198622525072705635801/388799208512064000000\) | \(714716220034934649000000000000\) | \([2, 2]\) | \(743178240\) | \(4.8388\) | |
| 169050.jc1 | 169050bz3 | \([1, 0, 0, -1030594113963, 402698032351746417]\) | \(65853432878493908038433301506521/38511703125000000\) | \(70794740014892578125000000\) | \([2]\) | \(1486356480\) | \(5.1854\) | |
| 169050.jc3 | 169050bz4 | \([1, 0, 0, -62011505963, 6782780414898417]\) | \(-14346048055032350809895395801/2509530875136386550792000\) | \(-4613184342639386583033250125000000\) | \([2]\) | \(1486356480\) | \(5.1854\) |
Rank
sage: E.rank()
The elliptic curves in class 169050bz have rank \(0\).
Complex multiplication
The elliptic curves in class 169050bz do not have complex multiplication.Modular form 169050.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
