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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 450840q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 450840.q4 | 450840q1 | \([0, -1, 0, -3037775, -414540048]\) | \(8027441608013824/4452347908125\) | \(1719501677509965570000\) | \([2]\) | \(28311552\) | \(2.7651\) | \(\Gamma_0(N)\)-optimal |
| 450840.q2 | 450840q2 | \([0, -1, 0, -36863780, -86007863100]\) | \(896581610757188944/1545359765625\) | \(9549114360984900000000\) | \([2, 2]\) | \(56623104\) | \(3.1117\) | |
| 450840.q3 | 450840q3 | \([0, -1, 0, -25367360, -140643449508]\) | \(-73039208963041156/303497314453125\) | \(-7501503865781250000000000\) | \([4]\) | \(113246208\) | \(3.4583\) | |
| 450840.q1 | 450840q4 | \([0, -1, 0, -589576280, -5509886168100]\) | \(916959671620739147236/2731145625\) | \(67505373155825280000\) | \([2]\) | \(113246208\) | \(3.4583\) |
Rank
sage: E.rank()
The elliptic curves in class 450840q have rank \(0\).
Complex multiplication
The elliptic curves in class 450840q do not have complex multiplication.Modular form 450840.2.a.q
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.
