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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 174570bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 174570.bt6 | 174570bf1 | \([1, 1, 1, 134884, -1755907]\) | \(1833318007919/1070530560\) | \(-158476943151267840\) | \([2]\) | \(2433024\) | \(1.9898\) | \(\Gamma_0(N)\)-optimal |
| 174570.bt5 | 174570bf2 | \([1, 1, 1, -542236, -14756611]\) | \(119102750067601/68309049600\) | \(10112190884281094400\) | \([2, 2]\) | \(4866048\) | \(2.3364\) | |
| 174570.bt3 | 174570bf3 | \([1, 1, 1, -5662956, 5163315453]\) | \(135670761487282321/643043610000\) | \(95193532472119290000\) | \([2, 2]\) | \(9732096\) | \(2.6829\) | |
| 174570.bt2 | 174570bf4 | \([1, 1, 1, -6255436, -6011331331]\) | \(182864522286982801/463015182960\) | \(68542864229981251440\) | \([2]\) | \(9732096\) | \(2.6829\) | |
| 174570.bt1 | 174570bf5 | \([1, 1, 1, -90503976, 331360069149]\) | \(553808571467029327441/12529687500\) | \(1854843427954687500\) | \([2]\) | \(19464192\) | \(3.0295\) | |
| 174570.bt4 | 174570bf6 | \([1, 1, 1, -2753456, 10465588253]\) | \(-15595206456730321/310672490129100\) | \(-45990678264105043269900\) | \([2]\) | \(19464192\) | \(3.0295\) |
Rank
sage: E.rank()
The elliptic curves in class 174570bf have rank \(1\).
Complex multiplication
The elliptic curves in class 174570bf do not have complex multiplication.Modular form 174570.2.a.bf
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
