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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 67830.bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67830.bz1 | 67830bz8 | \([1, 0, 0, -25603575, -42731601393]\) | \(1856203306931677398202594801/285442925181016286066250\) | \(285442925181016286066250\) | \([2]\) | \(8957952\) | \(3.2242\) | |
| 67830.bz2 | 67830bz6 | \([1, 0, 0, -24571845, -46882663875]\) | \(1640729605302312040170582481/50078778067225044900\) | \(50078778067225044900\) | \([2, 2]\) | \(4478976\) | \(2.8776\) | |
| 67830.bz3 | 67830bz3 | \([1, 0, 0, -24571665, -46883385063]\) | \(1640693548282750959454626961/1528553391120\) | \(1528553391120\) | \([2]\) | \(2239488\) | \(2.5311\) | |
| 67830.bz4 | 67830bz7 | \([1, 0, 0, -23542995, -50987569605]\) | \(-1443141263044885978311580081/287846712789197778248970\) | \(-287846712789197778248970\) | \([2]\) | \(8957952\) | \(3.2242\) | |
| 67830.bz5 | 67830bz5 | \([1, 0, 0, -6834825, 6870319857]\) | \(35310666410995026859894801/40072943900390625000\) | \(40072943900390625000\) | \([6]\) | \(2985984\) | \(2.6749\) | |
| 67830.bz6 | 67830bz2 | \([1, 0, 0, -537345, 47630025]\) | \(17158661194925340654481/8947893637809000000\) | \(8947893637809000000\) | \([2, 6]\) | \(1492992\) | \(2.3283\) | |
| 67830.bz7 | 67830bz1 | \([1, 0, 0, -304065, -64017783]\) | \(3109017019607132956561/30145442277888000\) | \(30145442277888000\) | \([6]\) | \(746496\) | \(1.9818\) | \(\Gamma_0(N)\)-optimal |
| 67830.bz8 | 67830bz4 | \([1, 0, 0, 2027655, 371333025]\) | \(921946855702725447905519/593047570085451873000\) | \(-593047570085451873000\) | \([6]\) | \(2985984\) | \(2.6749\) |
Rank
sage: E.rank()
The elliptic curves in class 67830.bz have rank \(0\).
Complex multiplication
The elliptic curves in class 67830.bz do not have complex multiplication.Modular form 67830.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
