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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 190575.bz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
190575.bz1 | 190575bh3 | \([1, -1, 1, -15654852005, 753917483251122]\) | \(21026497979043461623321/161783881875\) | \(3264667208469408779296875\) | \([2]\) | \(176947200\) | \(4.2963\) | |
190575.bz2 | 190575bh2 | \([1, -1, 1, -979079630, 11763674247372]\) | \(5143681768032498601/14238434358225\) | \(287320029724322799922265625\) | \([2, 2]\) | \(88473600\) | \(3.9497\) | |
190575.bz3 | 190575bh4 | \([1, -1, 1, -593165255, 21131359786122]\) | \(-1143792273008057401/8897444448004035\) | \(-179543195477407768519608046875\) | \([2]\) | \(176947200\) | \(4.2963\) | |
190575.bz4 | 190575bh1 | \([1, -1, 1, -85963505, 20983435872]\) | \(3481467828171481/2005331497785\) | \(40465959322126905447890625\) | \([2]\) | \(44236800\) | \(3.6031\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 190575.bz have rank \(0\).
Complex multiplication
The elliptic curves in class 190575.bz do not have complex multiplication.Modular form 190575.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.