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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 432450b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 432450.b3 | 432450b1 | \([1, -1, 0, -52655292, 147063983616]\) | \(1597099875769/186000\) | \(1880315220648656250000\) | \([2]\) | \(53084160\) | \(3.1085\) | \(\Gamma_0(N)\)-optimal |
| 432450.b2 | 432450b2 | \([1, -1, 0, -56979792, 121493215116]\) | \(2023804595449/540562500\) | \(5464666110010157226562500\) | \([2, 2]\) | \(106168320\) | \(3.4550\) | |
| 432450.b4 | 432450b3 | \([1, -1, 0, 144109458, 786897543366]\) | \(32740359775271/45410156250\) | \(-459061333166175842285156250\) | \([2]\) | \(212336640\) | \(3.8016\) | |
| 432450.b1 | 432450b4 | \([1, -1, 0, -327261042, -2180492191134]\) | \(383432500775449/18701300250\) | \(189055588741911399410156250\) | \([2]\) | \(212336640\) | \(3.8016\) |
Rank
sage: E.rank()
The elliptic curves in class 432450b have rank \(1\).
Complex multiplication
The elliptic curves in class 432450b do not have complex multiplication.Modular form 432450.2.a.b
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.
