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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 7728.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7728.o1 | 7728r5 | \([0, 1, 0, -1266144, 547941492]\) | \(54804145548726848737/637608031452\) | \(2611642496827392\) | \([4]\) | \(98304\) | \(2.1091\) | |
| 7728.o2 | 7728r3 | \([0, 1, 0, -283424, -58169868]\) | \(614716917569296417/19093020912\) | \(78205013655552\) | \([2]\) | \(49152\) | \(1.7625\) | |
| 7728.o3 | 7728r4 | \([0, 1, 0, -81184, 8073716]\) | \(14447092394873377/1439452851984\) | \(5895998881726464\) | \([2, 4]\) | \(49152\) | \(1.7625\) | |
| 7728.o4 | 7728r2 | \([0, 1, 0, -18464, -832524]\) | \(169967019783457/26337394944\) | \(107877969690624\) | \([2, 2]\) | \(24576\) | \(1.4160\) | |
| 7728.o5 | 7728r1 | \([0, 1, 0, 2016, -70668]\) | \(221115865823/664731648\) | \(-2722740830208\) | \([2]\) | \(12288\) | \(1.0694\) | \(\Gamma_0(N)\)-optimal |
| 7728.o6 | 7728r6 | \([0, 1, 0, 100256, 39208820]\) | \(27207619911317663/177609314617308\) | \(-727487752672493568\) | \([4]\) | \(98304\) | \(2.1091\) |
Rank
sage: E.rank()
The elliptic curves in class 7728.o have rank \(0\).
Complex multiplication
The elliptic curves in class 7728.o do not have complex multiplication.Modular form 7728.2.a.o
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
