Show commands:
SageMath
E = EllipticCurve("eo1")
E.isogeny_class()
Elliptic curves in class 29400.eo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
29400.eo1 | 29400br4 | \([0, 1, 0, -8232408, -9094305312]\) | \(32779037733124/315\) | \(592950960000000\) | \([2]\) | \(589824\) | \(2.4130\) | |
29400.eo2 | 29400br6 | \([0, 1, 0, -7938408, 8577152688]\) | \(14695548366242/57421875\) | \(216180037500000000000\) | \([2]\) | \(1179648\) | \(2.7596\) | |
29400.eo3 | 29400br3 | \([0, 1, 0, -735408, -8823312]\) | \(23366901604/13505625\) | \(25422772410000000000\) | \([2, 2]\) | \(589824\) | \(2.4130\) | |
29400.eo4 | 29400br2 | \([0, 1, 0, -514908, -142005312]\) | \(32082281296/99225\) | \(46694888100000000\) | \([2, 2]\) | \(294912\) | \(2.0664\) | |
29400.eo5 | 29400br1 | \([0, 1, 0, -18783, -4082562]\) | \(-24918016/229635\) | \(-6754082028750000\) | \([4]\) | \(147456\) | \(1.7199\) | \(\Gamma_0(N)\)-optimal |
29400.eo6 | 29400br5 | \([0, 1, 0, 2939592, -67623312]\) | \(746185003198/432360075\) | \(-1627735374837600000000\) | \([2]\) | \(1179648\) | \(2.7596\) |
Rank
sage: E.rank()
The elliptic curves in class 29400.eo have rank \(1\).
Complex multiplication
The elliptic curves in class 29400.eo do not have complex multiplication.Modular form 29400.2.a.eo
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.