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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 236691.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 236691.r1 | 236691r5 | \([1, -1, 0, -105546033, 417088014696]\) | \(7389727131216686257/6115533215337\) | \(107610682513644759203937\) | \([2]\) | \(28311552\) | \(3.3478\) | |
| 236691.r2 | 236691r3 | \([1, -1, 0, -8047548, 3440942235]\) | \(3275619238041697/1605271262049\) | \(28246815125688692964249\) | \([2, 2]\) | \(14155776\) | \(3.0012\) | |
| 236691.r3 | 236691r2 | \([1, -1, 0, -4289103, -3380635440]\) | \(495909170514577/6224736609\) | \(109532256857384806809\) | \([2, 2]\) | \(7077888\) | \(2.6547\) | |
| 236691.r4 | 236691r1 | \([1, -1, 0, -4276098, -3402382401]\) | \(491411892194497/78897\) | \(1388294318635497\) | \([2]\) | \(3538944\) | \(2.3081\) | \(\Gamma_0(N)\)-optimal |
| 236691.r5 | 236691r4 | \([1, -1, 0, -738738, -8810563671]\) | \(-2533811507137/1904381781393\) | \(-33510049908372294686793\) | \([2]\) | \(14155776\) | \(3.0012\) | |
| 236691.r6 | 236691r6 | \([1, -1, 0, 29315817, 26329739634]\) | \(158346567380527343/108665074944153\) | \(-1912101932634550190177553\) | \([2]\) | \(28311552\) | \(3.3478\) |
Rank
sage: E.rank()
The elliptic curves in class 236691.r have rank \(0\).
Complex multiplication
The elliptic curves in class 236691.r do not have complex multiplication.Modular form 236691.2.a.r
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
