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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 62400cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
62400.fy4 | 62400cr1 | \([0, 1, 0, -2308, -6862]\) | \(1360251712/771147\) | \(771147000000\) | \([2]\) | \(98304\) | \(0.97086\) | \(\Gamma_0(N)\)-optimal |
62400.fy2 | 62400cr2 | \([0, 1, 0, -23433, 1366263]\) | \(22235451328/123201\) | \(7884864000000\) | \([2, 2]\) | \(196608\) | \(1.3174\) | |
62400.fy3 | 62400cr3 | \([0, 1, 0, -10433, 2887263]\) | \(-245314376/6908733\) | \(-3537271296000000\) | \([2]\) | \(393216\) | \(1.6640\) | |
62400.fy1 | 62400cr4 | \([0, 1, 0, -374433, 88063263]\) | \(11339065490696/351\) | \(179712000000\) | \([2]\) | \(393216\) | \(1.6640\) |
Rank
sage: E.rank()
The elliptic curves in class 62400cr have rank \(1\).
Complex multiplication
The elliptic curves in class 62400cr do not have complex multiplication.Modular form 62400.2.a.cr
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.