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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 82110bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 82110.bs6 | 82110bt1 | \([1, 0, 0, -49095400, 132402324032]\) | \(13087181362848921857775657601/10116926999101440000\) | \(10116926999101440000\) | \([8]\) | \(7340032\) | \(2.9544\) | \(\Gamma_0(N)\)-optimal |
| 82110.bs5 | 82110bt2 | \([1, 0, 0, -49423080, 130545230400]\) | \(13350979617415439280823624321/363628103290905600000000\) | \(363628103290905600000000\) | \([2, 8]\) | \(14680064\) | \(3.3010\) | |
| 82110.bs7 | 82110bt3 | \([1, 0, 0, 10134040, 425388708672]\) | \(115099000398621243971890559/78239096930039062500000000\) | \(-78239096930039062500000000\) | \([8]\) | \(29360128\) | \(3.6475\) | |
| 82110.bs4 | 82110bt4 | \([1, 0, 0, -114223080, -283150929600]\) | \(164810665209657549410090824321/60743509039087274201760000\) | \(60743509039087274201760000\) | \([2, 4]\) | \(29360128\) | \(3.6475\) | |
| 82110.bs8 | 82110bt5 | \([1, 0, 0, 352354920, -2007529902000]\) | \(4837987390362436347081585367679/4540848316592979232425603600\) | \(-4540848316592979232425603600\) | \([4]\) | \(58720256\) | \(3.9941\) | |
| 82110.bs2 | 82110bt6 | \([1, 0, 0, -1617601080, -25035066997200]\) | \(468099305477291219804418298216321/135739171637240733141123600\) | \(135739171637240733141123600\) | \([2, 2]\) | \(58720256\) | \(3.9941\) | |
| 82110.bs3 | 82110bt7 | \([1, 0, 0, -1409434380, -31714678433460]\) | \(-309640881349964101700603056547521/255321054523588348291733959740\) | \(-255321054523588348291733959740\) | \([2]\) | \(117440512\) | \(4.3407\) | |
| 82110.bs1 | 82110bt8 | \([1, 0, 0, -25879815780, -1602472070375340]\) | \(1916934412547006969354120058646317121/161260899317970551066940\) | \(161260899317970551066940\) | \([2]\) | \(117440512\) | \(4.3407\) |
Rank
sage: E.rank()
The elliptic curves in class 82110bt have rank \(1\).
Complex multiplication
The elliptic curves in class 82110bt do not have complex multiplication.Modular form 82110.2.a.bt
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
