Show commands:
SageMath
E = EllipticCurve("hp1")
E.isogeny_class()
Elliptic curves in class 224400.hp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 224400.hp1 | 224400cs4 | \([0, 1, 0, -88145408, 316059211188]\) | \(1183430669265454849849/10449720703125000\) | \(668782125000000000000000\) | \([2]\) | \(39813120\) | \(3.3951\) | |
| 224400.hp2 | 224400cs3 | \([0, 1, 0, -9537408, -3246484812]\) | \(1499114720492202169/796539777000000\) | \(50978545728000000000000\) | \([2]\) | \(19906560\) | \(3.0486\) | |
| 224400.hp3 | 224400cs2 | \([0, 1, 0, -7559408, -7744048812]\) | \(746461053445307689/27443694341250\) | \(1756396437840000000000\) | \([2]\) | \(13271040\) | \(2.8458\) | |
| 224400.hp4 | 224400cs1 | \([0, 1, 0, -7491408, -7894600812]\) | \(726497538898787209/1038579300\) | \(66469075200000000\) | \([2]\) | \(6635520\) | \(2.4992\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 224400.hp have rank \(0\).
Complex multiplication
The elliptic curves in class 224400.hp do not have complex multiplication.Modular form 224400.2.a.hp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
