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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 372645bl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 372645.bl6 | 372645bl1 | \([1, -1, 1, -8199743, 9039273702]\) | \(147281603041/5265\) | \(2179587245144985585\) | \([2]\) | \(12386304\) | \(2.6079\) | \(\Gamma_0(N)\)-optimal |
| 372645.bl5 | 372645bl2 | \([1, -1, 1, -8572388, 8172948606]\) | \(168288035761/27720225\) | \(11475526845688349105025\) | \([2, 2]\) | \(24772608\) | \(2.9544\) | |
| 372645.bl7 | 372645bl3 | \([1, -1, 1, 15649537, 45968840376]\) | \(1023887723039/2798036865\) | \(-1158322025147096284277985\) | \([2]\) | \(49545216\) | \(3.3010\) | |
| 372645.bl4 | 372645bl4 | \([1, -1, 1, -38756633, -85072221048]\) | \(15551989015681/1445900625\) | \(598569147197941666280625\) | \([2, 2]\) | \(49545216\) | \(3.3010\) | |
| 372645.bl8 | 372645bl5 | \([1, -1, 1, 45088492, -402878782848]\) | \(24487529386319/183539412225\) | \(-75981037391646275247293025\) | \([2]\) | \(99090432\) | \(3.6476\) | |
| 372645.bl2 | 372645bl6 | \([1, -1, 1, -605549678, -5735318728044]\) | \(59319456301170001/594140625\) | \(245960366205597331640625\) | \([2, 2]\) | \(99090432\) | \(3.6476\) | |
| 372645.bl3 | 372645bl7 | \([1, -1, 1, -591016523, -6023697216078]\) | \(-55150149867714721/5950927734375\) | \(-2463545334591319427490234375\) | \([2]\) | \(198180864\) | \(3.9942\) | |
| 372645.bl1 | 372645bl8 | \([1, -1, 1, -9688771553, -367069518204294]\) | \(242970740812818720001/24375\) | \(10090681690486044375\) | \([2]\) | \(198180864\) | \(3.9942\) |
Rank
sage: E.rank()
The elliptic curves in class 372645bl have rank \(1\).
Complex multiplication
The elliptic curves in class 372645bl do not have complex multiplication.Modular form 372645.2.a.bl
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.
