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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 169065.bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169065.bi1 | 169065x7 | \([1, -1, 0, -338130054, 2393255770753]\) | \(242970740812818720001/24375\) | \(428909515149375\) | \([2]\) | \(15728640\) | \(3.1553\) | |
| 169065.bi2 | 169065x5 | \([1, -1, 0, -21133179, 37398395128]\) | \(59319456301170001/594140625\) | \(10454669431766015625\) | \([2, 2]\) | \(7864320\) | \(2.8088\) | |
| 169065.bi3 | 169065x8 | \([1, -1, 0, -20625984, 39278364115]\) | \(-55150149867714721/5950927734375\) | \(-104714237097015380859375\) | \([2]\) | \(15728640\) | \(3.1553\) | |
| 169065.bi4 | 169065x3 | \([1, -1, 0, -1352574, 555040255]\) | \(15551989015681/1445900625\) | \(25442483529145775625\) | \([2, 2]\) | \(3932160\) | \(2.4622\) | |
| 169065.bi5 | 169065x2 | \([1, -1, 0, -299169, -53195792]\) | \(168288035761/27720225\) | \(487773057008475225\) | \([2, 2]\) | \(1966080\) | \(2.1156\) | |
| 169065.bi6 | 169065x1 | \([1, -1, 0, -286164, -58847765]\) | \(147281603041/5265\) | \(92644455272265\) | \([2]\) | \(983040\) | \(1.7690\) | \(\Gamma_0(N)\)-optimal |
| 169065.bi7 | 169065x4 | \([1, -1, 0, 546156, -299861627]\) | \(1023887723039/2798036865\) | \(-49235061954347783865\) | \([2]\) | \(3932160\) | \(2.4622\) | |
| 169065.bi8 | 169065x6 | \([1, -1, 0, 1573551, 2626151530]\) | \(24487529386319/183539412225\) | \(-3229612320337477767225\) | \([2]\) | \(7864320\) | \(2.8088\) |
Rank
sage: E.rank()
The elliptic curves in class 169065.bi have rank \(0\).
Complex multiplication
The elliptic curves in class 169065.bi do not have complex multiplication.Modular form 169065.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
