Show commands:
SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 198744.cj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
198744.cj1 | 198744i3 | \([0, 1, 0, -2520184, 1501333952]\) | \(3044193988/85293\) | \(49597718645077005312\) | \([2]\) | \(6193152\) | \(2.5583\) | |
198744.cj2 | 198744i2 | \([0, 1, 0, -367124, -52314144]\) | \(37642192/13689\) | \(1990031920944447744\) | \([2, 2]\) | \(3096576\) | \(2.2117\) | |
198744.cj3 | 198744i1 | \([0, 1, 0, -325719, -71641998]\) | \(420616192/117\) | \(1063051239820752\) | \([2]\) | \(1548288\) | \(1.8651\) | \(\Gamma_0(N)\)-optimal |
198744.cj4 | 198744i4 | \([0, 1, 0, 1123456, -368317104]\) | \(269676572/257049\) | \(-149473508728716297216\) | \([2]\) | \(6193152\) | \(2.5583\) |
Rank
sage: E.rank()
The elliptic curves in class 198744.cj have rank \(0\).
Complex multiplication
The elliptic curves in class 198744.cj do not have complex multiplication.Modular form 198744.2.a.cj
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.