Show commands:
SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 25200bl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25200.ep4 | 25200bl1 | \([0, 0, 0, 1950, -273625]\) | \(4499456/180075\) | \(-32818668750000\) | \([2]\) | \(49152\) | \(1.2734\) | \(\Gamma_0(N)\)-optimal |
25200.ep3 | 25200bl2 | \([0, 0, 0, -53175, -4518250]\) | \(5702413264/275625\) | \(803722500000000\) | \([2, 2]\) | \(98304\) | \(1.6199\) | |
25200.ep2 | 25200bl3 | \([0, 0, 0, -147675, 15988250]\) | \(30534944836/8203125\) | \(95681250000000000\) | \([2]\) | \(196608\) | \(1.9665\) | |
25200.ep1 | 25200bl4 | \([0, 0, 0, -840675, -296680750]\) | \(5633270409316/14175\) | \(165337200000000\) | \([2]\) | \(196608\) | \(1.9665\) |
Rank
sage: E.rank()
The elliptic curves in class 25200bl have rank \(1\).
Complex multiplication
The elliptic curves in class 25200bl do not have complex multiplication.Modular form 25200.2.a.bl
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.