Show commands:
SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 95550bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 95550.m5 | 95550bt1 | \([1, 1, 0, -453275, -1850731875]\) | \(-5602762882081/801531494400\) | \(-1473427793510400000000\) | \([2]\) | \(7077888\) | \(2.7412\) | \(\Gamma_0(N)\)-optimal |
| 95550.m4 | 95550bt2 | \([1, 1, 0, -25541275, -49292139875]\) | \(1002404925316922401/9348917760000\) | \(17185794149160000000000\) | \([2, 2]\) | \(14155776\) | \(3.0878\) | |
| 95550.m3 | 95550bt3 | \([1, 1, 0, -44749275, 34858108125]\) | \(5391051390768345121/2833965225000000\) | \(5209580855562890625000000\) | \([2, 2]\) | \(28311552\) | \(3.4344\) | |
| 95550.m2 | 95550bt4 | \([1, 1, 0, -407741275, -3169190739875]\) | \(4078208988807294650401/359723582400\) | \(661267496027775000000\) | \([2]\) | \(28311552\) | \(3.4344\) | |
| 95550.m6 | 95550bt5 | \([1, 1, 0, 169625725, 272171233125]\) | \(293623352309352854879/187320324116835000\) | \(-344344512687836264296875000\) | \([2]\) | \(56623104\) | \(3.7809\) | |
| 95550.m1 | 95550bt6 | \([1, 1, 0, -566452275, 5183545015125]\) | \(10934663514379917006241/12996826171875000\) | \(23891618785858154296875000\) | \([2]\) | \(56623104\) | \(3.7809\) |
Rank
sage: E.rank()
The elliptic curves in class 95550bt have rank \(1\).
Complex multiplication
The elliptic curves in class 95550bt do not have complex multiplication.Modular form 95550.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
