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SageMath
E = EllipticCurve("qh1")
E.isogeny_class()
Elliptic curves in class 176400qh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 176400.rm5 | 176400qh1 | \([0, 0, 0, -169050, -110060125]\) | \(-24918016/229635\) | \(-4923725798958750000\) | \([2]\) | \(2359296\) | \(2.2692\) | \(\Gamma_0(N)\)-optimal |
| 176400.rm4 | 176400qh2 | \([0, 0, 0, -4634175, -3829509250]\) | \(32082281296/99225\) | \(34040573424900000000\) | \([2, 2]\) | \(4718592\) | \(2.6157\) | |
| 176400.rm3 | 176400qh3 | \([0, 0, 0, -6618675, -231610750]\) | \(23366901604/13505625\) | \(18533201086890000000000\) | \([2, 2]\) | \(9437184\) | \(2.9623\) | |
| 176400.rm1 | 176400qh4 | \([0, 0, 0, -74091675, -245472151750]\) | \(32779037733124/315\) | \(432261249840000000\) | \([2]\) | \(9437184\) | \(2.9623\) | |
| 176400.rm2 | 176400qh5 | \([0, 0, 0, -71445675, 231654568250]\) | \(14695548366242/57421875\) | \(157595247337500000000000\) | \([2]\) | \(18874368\) | \(3.3089\) | |
| 176400.rm6 | 176400qh6 | \([0, 0, 0, 26456325, -1852285750]\) | \(746185003198/432360075\) | \(-1186619088256610400000000\) | \([2]\) | \(18874368\) | \(3.3089\) |
Rank
sage: E.rank()
The elliptic curves in class 176400qh have rank \(1\).
Complex multiplication
The elliptic curves in class 176400qh do not have complex multiplication.Modular form 176400.2.a.qh
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
