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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 54600.bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54600.bt1 | 54600cd4 | \([0, 1, 0, -42575408, 87324086688]\) | \(266716694084614489298/51372277695070605\) | \(1643912886242259360000000\) | \([2]\) | \(8257536\) | \(3.3638\) | |
| 54600.bt2 | 54600cd2 | \([0, 1, 0, -40370408, 98710706688]\) | \(454771411897393003396/23468066028225\) | \(375489056451600000000\) | \([2, 2]\) | \(4128768\) | \(3.0172\) | |
| 54600.bt3 | 54600cd1 | \([0, 1, 0, -40369908, 98713274688]\) | \(1819018058610682173904/4844385\) | \(19377540000000\) | \([4]\) | \(2064384\) | \(2.6706\) | \(\Gamma_0(N)\)-optimal |
| 54600.bt4 | 54600cd3 | \([0, 1, 0, -38173408, 109932982688]\) | \(-192245661431796830258/51935513760073125\) | \(-1661936440322340000000000\) | \([2]\) | \(8257536\) | \(3.3638\) |
Rank
sage: E.rank()
The elliptic curves in class 54600.bt have rank \(0\).
Complex multiplication
The elliptic curves in class 54600.bt do not have complex multiplication.Modular form 54600.2.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 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.
