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SageMath
E = EllipticCurve("ky1")
E.isogeny_class()
Elliptic curves in class 463680.ky
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.ky1 | 463680ky4 | \([0, 0, 0, -277839372, 1744518730736]\) | \(12411881707829361287041/303132494474220600\) | \(57929521816327111940505600\) | \([2]\) | \(191102976\) | \(3.7275\) | \(\Gamma_0(N)\)-optimal* |
463680.ky2 | 463680ky2 | \([0, 0, 0, -34191372, -76021198864]\) | \(23131609187144855041/322060536000000\) | \(61546726881755136000000\) | \([2]\) | \(63700992\) | \(3.1782\) | \(\Gamma_0(N)\)-optimal* |
463680.ky3 | 463680ky1 | \([0, 0, 0, -276492, -3185602576]\) | \(-12232183057921/22933241856000\) | \(-4382610768009363456000\) | \([2]\) | \(31850496\) | \(2.8316\) | \(\Gamma_0(N)\)-optimal* |
463680.ky4 | 463680ky3 | \([0, 0, 0, 2488308, 85988044784]\) | \(8915971454369279/16719623332762560\) | \(-3195169776489963517378560\) | \([2]\) | \(95551488\) | \(3.3809\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 463680.ky have rank \(1\).
Complex multiplication
The elliptic curves in class 463680.ky do not have complex multiplication.Modular form 463680.2.a.ky
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.