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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 78030.bz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
78030.bz1 | 78030cg2 | \([1, -1, 1, -414047, -102464729]\) | \(-16522921323/4000\) | \(-1900399082508000\) | \([]\) | \(907200\) | \(1.9205\) | |
78030.bz2 | 78030cg1 | \([1, -1, 1, 2113, -496281]\) | \(1601613/163840\) | \(-106776881233920\) | \([]\) | \(302400\) | \(1.3712\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 78030.bz have rank \(0\).
Complex multiplication
The elliptic curves in class 78030.bz do not have complex multiplication.Modular form 78030.2.a.bz
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.