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SageMath
E = EllipticCurve("en1")
E.isogeny_class()
Elliptic curves in class 274890en
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 274890.en7 | 274890en1 | \([1, 1, 1, -11507655685, -403142098010485]\) | \(1432504679512464302827718035009/232233326153721446400000000\) | \(27322018588659174447513600000000\) | \([2]\) | \(891813888\) | \(4.7541\) | \(\Gamma_0(N)\)-optimal |
| 274890.en6 | 274890en2 | \([1, 1, 1, -51648455685, 4131756753829515]\) | \(129511249478743944259581330835009/12262789317997149185802240000\) | \(1442704900473046604560447733760000\) | \([2, 2]\) | \(1783627776\) | \(5.1007\) | |
| 274890.en5 | 274890en3 | \([1, 1, 1, -255275709445, 49599171128647307]\) | \(15637378471582822120727563649467969/16113547119140625000000000000\) | \(1895742705019775390625000000000000\) | \([2]\) | \(2675441664\) | \(5.3034\) | |
| 274890.en4 | 274890en4 | \([1, 1, 1, -806599687685, 278824386282943115]\) | \(493298302018650738343048153196947009/5139490792463830279120089600\) | \(604655952242577168508199421350400\) | \([2]\) | \(3567255552\) | \(5.4473\) | |
| 274890.en8 | 274890en5 | \([1, 1, 1, 61049976315, 19673456558475915]\) | \(213890734289719241265598586476991/1544981081981970035652027609600\) | \(-181765479314096792724425396241830400\) | \([2]\) | \(3567255552\) | \(5.4473\) | |
| 274890.en2 | 274890en6 | \([1, 1, 1, -4083400709445, 3176002827378647307]\) | \(64003168104546012500462338813649467969/68064746081030015625000000\) | \(8007749311687100308265625000000\) | \([2, 2]\) | \(5350883328\) | \(5.6500\) | |
| 274890.en1 | 274890en7 | \([1, 1, 1, -65334411334445, 203264262729193897307]\) | \(262156976355489363181342849900999019467969/296485141924125000\) | \(34881180462231382125000\) | \([2]\) | \(10701766656\) | \(5.9966\) | |
| 274890.en3 | 274890en8 | \([1, 1, 1, -4082390084445, 3177653488063397307]\) | \(-63955658296770964115513956628279467969/66004356107812185925891924125000\) | \(-7765346491727995861995258981382125000\) | \([2]\) | \(10701766656\) | \(5.9966\) |
Rank
sage: E.rank()
The elliptic curves in class 274890en have rank \(1\).
Complex multiplication
The elliptic curves in class 274890en do not have complex multiplication.Modular form 274890.2.a.en
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
