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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 2366j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2366.j5 | 2366j1 | \([1, 0, 0, -88, 636]\) | \(-15625/28\) | \(-135150652\) | \([2]\) | \(720\) | \(0.25039\) | \(\Gamma_0(N)\)-optimal |
| 2366.j4 | 2366j2 | \([1, 0, 0, -1778, 28690]\) | \(128787625/98\) | \(473027282\) | \([2]\) | \(1440\) | \(0.59696\) | |
| 2366.j6 | 2366j3 | \([1, 0, 0, 757, -13391]\) | \(9938375/21952\) | \(-105958111168\) | \([2]\) | \(2160\) | \(0.79970\) | |
| 2366.j3 | 2366j4 | \([1, 0, 0, -6003, -147239]\) | \(4956477625/941192\) | \(4542954016328\) | \([2]\) | \(4320\) | \(1.1463\) | |
| 2366.j2 | 2366j5 | \([1, 0, 0, -28818, -1890812]\) | \(-548347731625/1835008\) | \(-8857233129472\) | \([2]\) | \(6480\) | \(1.3490\) | |
| 2366.j1 | 2366j6 | \([1, 0, 0, -461458, -120693756]\) | \(2251439055699625/25088\) | \(121094984192\) | \([2]\) | \(12960\) | \(1.6956\) |
Rank
sage: E.rank()
The elliptic curves in class 2366j have rank \(0\).
Complex multiplication
The elliptic curves in class 2366j do not have complex multiplication.Modular form 2366.2.a.j
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
