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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 61710bz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
61710.by3 | 61710bz1 | \([1, 1, 1, -6337680, 6137309025]\) | \(15891267085572193561/3334993530000\) | \(5908144473000330000\) | \([4]\) | \(2211840\) | \(2.5979\) | \(\Gamma_0(N)\)-optimal |
61710.by2 | 61710bz2 | \([1, 1, 1, -7037060, 4698264737]\) | \(21754112339458491481/7199734626562500\) | \(12754769074767689062500\) | \([2, 2]\) | \(4423680\) | \(2.9445\) | |
61710.by4 | 61710bz3 | \([1, 1, 1, 20341610, 32361672905]\) | \(525440531549759128199/559322204589843750\) | \(-990873404085388183593750\) | \([2]\) | \(8847360\) | \(3.2911\) | |
61710.by1 | 61710bz4 | \([1, 1, 1, -45605810, -115049990263]\) | \(5921450764096952391481/200074809015963750\) | \(354444728735129756913750\) | \([2]\) | \(8847360\) | \(3.2911\) |
Rank
sage: E.rank()
The elliptic curves in class 61710bz have rank \(1\).
Complex multiplication
The elliptic curves in class 61710bz do not have complex multiplication.Modular form 61710.2.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 Cremona 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 Cremona labels.