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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 180336bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 180336.bz3 | 180336bz1 | \([0, 1, 0, -2119, -22360]\) | \(2725888/1053\) | \(406669762512\) | \([2]\) | \(163840\) | \(0.92659\) | \(\Gamma_0(N)\)-optimal |
| 180336.bz2 | 180336bz2 | \([0, 1, 0, -15124, 695516]\) | \(61918288/1521\) | \(9398590066944\) | \([2, 2]\) | \(327680\) | \(1.2732\) | |
| 180336.bz1 | 180336bz3 | \([0, 1, 0, -240544, 45328676]\) | \(62275269892/39\) | \(963957955584\) | \([2]\) | \(655360\) | \(1.6197\) | |
| 180336.bz4 | 180336bz4 | \([0, 1, 0, 2216, 2214500]\) | \(48668/85683\) | \(-2117815628418048\) | \([2]\) | \(655360\) | \(1.6197\) |
Rank
sage: E.rank()
The elliptic curves in class 180336bz have rank \(1\).
Complex multiplication
The elliptic curves in class 180336bz do not have complex multiplication.Modular form 180336.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
