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SageMath
E = EllipticCurve("bu1")
E.isogeny_class()
Elliptic curves in class 117810.bu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 117810.bu1 | 117810bt8 | \([1, -1, 0, -12000198000204, 16000399002091373928]\) | \(262156976355489363181342849900999019467969/296485141924125000\) | \(216137668462687125000\) | \([6]\) | \(1783627776\) | \(5.5729\) | |
| 117810.bu2 | 117810bt6 | \([1, -1, 0, -750012375204, 250006374900998928]\) | \(64003168104546012500462338813649467969/68064746081030015625000000\) | \(49619199893070881390625000000\) | \([2, 6]\) | \(891813888\) | \(5.2263\) | |
| 117810.bu3 | 117810bt7 | \([1, -1, 0, -749826750204, 250136310210623928]\) | \(-63955658296770964115513956628279467969/66004356107812185925891924125000\) | \(-48117175602595083539975212687125000\) | \([6]\) | \(1783627776\) | \(5.5729\) | |
| 117810.bu4 | 117810bt5 | \([1, -1, 0, -148150963044, 21948338798341200]\) | \(493298302018650738343048153196947009/5139490792463830279120089600\) | \(3746688787706132273478545318400\) | \([2]\) | \(594542592\) | \(5.0236\) | |
| 117810.bu5 | 117810bt3 | \([1, -1, 0, -46887375204, 3904328025998928]\) | \(15637378471582822120727563649467969/16113547119140625000000000000\) | \(11746775849853515625000000000000\) | \([6]\) | \(445906944\) | \(4.8798\) | |
| 117810.bu6 | 117810bt2 | \([1, -1, 0, -9486451044, 325244393182800]\) | \(129511249478743944259581330835009/12262789317997149185802240000\) | \(8939573412819921756449832960000\) | \([2, 2]\) | \(297271296\) | \(4.6770\) | |
| 117810.bu7 | 117810bt1 | \([1, -1, 0, -2113651044, -31733311777200]\) | \(1432504679512464302827718035009/232233326153721446400000000\) | \(169298094766062934425600000000\) | \([2]\) | \(148635648\) | \(4.3305\) | \(\Gamma_0(N)\)-optimal |
| 117810.bu8 | 117810bt4 | \([1, -1, 0, 11213260956, 1548634631864400]\) | \(213890734289719241265598586476991/1544981081981970035652027609600\) | \(-1126291208764856155990328127398400\) | \([2]\) | \(594542592\) | \(5.0236\) |
Rank
sage: E.rank()
The elliptic curves in class 117810.bu have rank \(0\).
Complex multiplication
The elliptic curves in class 117810.bu do not have complex multiplication.Modular form 117810.2.a.bu
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
