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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 333270.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.m1 | 333270m3 | \([1, -1, 0, -8050950, -8714672064]\) | \(534774372149809/5323062500\) | \(574455126965355562500\) | \([2]\) | \(21897216\) | \(2.8020\) | |
333270.m2 | 333270m4 | \([1, -1, 0, -2099700, -21315848814]\) | \(-9486391169809/1813439640250\) | \(-195703074834049471610250\) | \([2]\) | \(43794432\) | \(3.1485\) | |
333270.m3 | 333270m1 | \([1, -1, 0, -719010, 228028500]\) | \(380920459249/12622400\) | \(1362186221673614400\) | \([2]\) | \(7299072\) | \(2.2527\) | \(\Gamma_0(N)\)-optimal |
333270.m4 | 333270m2 | \([1, -1, 0, 233190, 787350780]\) | \(12994449551/2489452840\) | \(-268657177569578600040\) | \([2]\) | \(14598144\) | \(2.5992\) |
Rank
sage: E.rank()
The elliptic curves in class 333270.m have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.m do not have complex multiplication.Modular form 333270.2.a.m
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.