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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 86190r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
86190.t3 | 86190r1 | \([1, 1, 0, -17072, 742656]\) | \(114013572049/15667200\) | \(75622581964800\) | \([2]\) | \(442368\) | \(1.3899\) | \(\Gamma_0(N)\)-optimal |
86190.t2 | 86190r2 | \([1, 1, 0, -71152, -6579776]\) | \(8253429989329/936360000\) | \(4519630875240000\) | \([2, 2]\) | \(884736\) | \(1.7364\) | |
86190.t4 | 86190r3 | \([1, 1, 0, 97848, -32909976]\) | \(21464092074671/109596256200\) | \(-529000195792465800\) | \([2]\) | \(1769472\) | \(2.0830\) | |
86190.t1 | 86190r4 | \([1, 1, 0, -1105432, -447803624]\) | \(30949975477232209/478125000\) | \(2307818053125000\) | \([2]\) | \(1769472\) | \(2.0830\) |
Rank
sage: E.rank()
The elliptic curves in class 86190r have rank \(1\).
Complex multiplication
The elliptic curves in class 86190r do not have complex multiplication.Modular form 86190.2.a.r
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.