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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 72450bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.bm3 | 72450bt1 | \([1, -1, 0, -79542, 8654116]\) | \(4886171981209/270480\) | \(3080936250000\) | \([2]\) | \(294912\) | \(1.4619\) | \(\Gamma_0(N)\)-optimal |
| 72450.bm2 | 72450bt2 | \([1, -1, 0, -84042, 7623616]\) | \(5763259856089/1143116100\) | \(13020806826562500\) | \([2, 2]\) | \(589824\) | \(1.8085\) | |
| 72450.bm4 | 72450bt3 | \([1, -1, 0, 174708, 45142366]\) | \(51774168853511/107398242630\) | \(-1223333107457343750\) | \([2]\) | \(1179648\) | \(2.1550\) | |
| 72450.bm1 | 72450bt4 | \([1, -1, 0, -414792, -95901134]\) | \(692895692874169/51420783750\) | \(585714864902343750\) | \([2]\) | \(1179648\) | \(2.1550\) |
Rank
sage: E.rank()
The elliptic curves in class 72450bt have rank \(2\).
Complex multiplication
The elliptic curves in class 72450bt do not have complex multiplication.Modular form 72450.2.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 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.
