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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 166635.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
166635.m1 | 166635o3 | \([1, -1, 1, -20441453, 35577721496]\) | \(8753151307882969/65205\) | \(7036803823696605\) | \([2]\) | \(5947392\) | \(2.6354\) | |
166635.m2 | 166635o2 | \([1, -1, 1, -1278428, 555377006]\) | \(2141202151369/5832225\) | \(629403008675085225\) | \([2, 2]\) | \(2973696\) | \(2.2888\) | |
166635.m3 | 166635o4 | \([1, -1, 1, -778523, 993893672]\) | \(-483551781049/3672913125\) | \(-396374037606095338125\) | \([2]\) | \(5947392\) | \(2.6354\) | |
166635.m4 | 166635o1 | \([1, -1, 1, -111983, 1082342]\) | \(1439069689/828345\) | \(89393470797330945\) | \([2]\) | \(1486848\) | \(1.9422\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 166635.m have rank \(1\).
Complex multiplication
The elliptic curves in class 166635.m do not have complex multiplication.Modular form 166635.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.