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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 164934.dz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
164934.dz1 | 164934bp2 | \([1, -1, 1, -1316174, -579377347]\) | \(2940001530995593/8673562656\) | \(743897824255577376\) | \([2]\) | \(2764800\) | \(2.2992\) | |
164934.dz2 | 164934bp1 | \([1, -1, 1, -116654, -728899]\) | \(2046931732873/1181672448\) | \(101347462157534208\) | \([2]\) | \(1382400\) | \(1.9526\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 164934.dz have rank \(1\).
Complex multiplication
The elliptic curves in class 164934.dz do not have complex multiplication.Modular form 164934.2.a.dz
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.