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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 27225.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27225.bp1 | 27225bi8 | \([1, -1, 0, -58806567, -173560171284]\) | \(1114544804970241/405\) | \(8172570741328125\) | \([2]\) | \(983040\) | \(2.8438\) | |
| 27225.bp2 | 27225bi6 | \([1, -1, 0, -3675942, -2710364409]\) | \(272223782641/164025\) | \(3309891150237890625\) | \([2, 2]\) | \(491520\) | \(2.4973\) | |
| 27225.bp3 | 27225bi7 | \([1, -1, 0, -2995317, -3745595034]\) | \(-147281603041/215233605\) | \(-4343239167342160078125\) | \([2]\) | \(983040\) | \(2.8438\) | |
| 27225.bp4 | 27225bi4 | \([1, -1, 0, -2178567, 1238213466]\) | \(56667352321/15\) | \(302687805234375\) | \([2]\) | \(245760\) | \(2.1507\) | |
| 27225.bp5 | 27225bi3 | \([1, -1, 0, -272817, -25298784]\) | \(111284641/50625\) | \(1021571342666015625\) | \([2, 2]\) | \(245760\) | \(2.1507\) | |
| 27225.bp6 | 27225bi2 | \([1, -1, 0, -136692, 19214091]\) | \(13997521/225\) | \(4540317078515625\) | \([2, 2]\) | \(122880\) | \(1.8041\) | |
| 27225.bp7 | 27225bi1 | \([1, -1, 0, -567, 837216]\) | \(-1/15\) | \(-302687805234375\) | \([2]\) | \(61440\) | \(1.4575\) | \(\Gamma_0(N)\)-optimal |
| 27225.bp8 | 27225bi5 | \([1, -1, 0, 952308, -190690659]\) | \(4733169839/3515625\) | \(-70942454351806640625\) | \([2]\) | \(491520\) | \(2.4973\) |
Rank
sage: E.rank()
The elliptic curves in class 27225.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 27225.bp do not have complex multiplication.Modular form 27225.2.a.bp
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
