Show commands:
SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 253920bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 253920.bz3 | 253920bz1 | \([0, 1, 0, -5466, -140616]\) | \(1906624/225\) | \(2131716801600\) | \([2, 2]\) | \(360448\) | \(1.0970\) | \(\Gamma_0(N)\)-optimal |
| 253920.bz2 | 253920bz2 | \([0, 1, 0, -21336, 1046460]\) | \(14172488/1875\) | \(142114453440000\) | \([2]\) | \(720896\) | \(1.4436\) | |
| 253920.bz4 | 253920bz3 | \([0, 1, 0, 7759, -704001]\) | \(85184/405\) | \(-245573775544320\) | \([2]\) | \(720896\) | \(1.4436\) | |
| 253920.bz1 | 253920bz4 | \([0, 1, 0, -84816, -9535656]\) | \(890277128/15\) | \(1136915627520\) | \([2]\) | \(720896\) | \(1.4436\) |
Rank
sage: E.rank()
The elliptic curves in class 253920bz have rank \(2\).
Complex multiplication
The elliptic curves in class 253920bz do not have complex multiplication.Modular form 253920.2.a.bz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
