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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 27456.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27456.z1 | 27456p4 | \([0, -1, 0, -439297, 112215553]\) | \(35765103905346817/1287\) | \(337379328\) | \([2]\) | \(131072\) | \(1.5808\) | |
| 27456.z2 | 27456p6 | \([0, -1, 0, -192577, -31433663]\) | \(3013001140430737/108679952667\) | \(28489797511938048\) | \([2]\) | \(262144\) | \(1.9274\) | |
| 27456.z3 | 27456p3 | \([0, -1, 0, -30337, 1371265]\) | \(11779205551777/3763454409\) | \(986566992592896\) | \([2, 2]\) | \(131072\) | \(1.5808\) | |
| 27456.z4 | 27456p2 | \([0, -1, 0, -27457, 1760065]\) | \(8732907467857/1656369\) | \(434207195136\) | \([2, 2]\) | \(65536\) | \(1.2342\) | |
| 27456.z5 | 27456p1 | \([0, -1, 0, -1537, 33793]\) | \(-1532808577/938223\) | \(-245949530112\) | \([2]\) | \(32768\) | \(0.88767\) | \(\Gamma_0(N)\)-optimal |
| 27456.z6 | 27456p5 | \([0, -1, 0, 85823, 9246913]\) | \(266679605718863/296110251723\) | \(-77623525827674112\) | \([2]\) | \(262144\) | \(1.9274\) |
Rank
sage: E.rank()
The elliptic curves in class 27456.z have rank \(0\).
Complex multiplication
The elliptic curves in class 27456.z do not have complex multiplication.Modular form 27456.2.a.z
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
