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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 22050bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.ba5 | 22050bi1 | \([1, -1, 0, -5742, 340416]\) | \(-15625/28\) | \(-37522677937500\) | \([2]\) | \(55296\) | \(1.2949\) | \(\Gamma_0(N)\)-optimal |
| 22050.ba4 | 22050bi2 | \([1, -1, 0, -115992, 15224166]\) | \(128787625/98\) | \(131329372781250\) | \([2]\) | \(110592\) | \(1.6415\) | |
| 22050.ba6 | 22050bi3 | \([1, -1, 0, 49383, -7101459]\) | \(9938375/21952\) | \(-29417779503000000\) | \([2]\) | \(165888\) | \(1.8442\) | |
| 22050.ba3 | 22050bi4 | \([1, -1, 0, -391617, -77220459]\) | \(4956477625/941192\) | \(1261287296191125000\) | \([2]\) | \(331776\) | \(2.1908\) | |
| 22050.ba2 | 22050bi5 | \([1, -1, 0, -1879992, -994555584]\) | \(-548347731625/1835008\) | \(-2459086221312000000\) | \([2]\) | \(497664\) | \(2.3935\) | |
| 22050.ba1 | 22050bi6 | \([1, -1, 0, -30103992, -63567163584]\) | \(2251439055699625/25088\) | \(33620319432000000\) | \([2]\) | \(995328\) | \(2.7401\) |
Rank
sage: E.rank()
The elliptic curves in class 22050bi have rank \(1\).
Complex multiplication
The elliptic curves in class 22050bi do not have complex multiplication.Modular form 22050.2.a.bi
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
