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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 12480x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12480.cm4 | 12480x1 | \([0, 1, 0, -81, 975]\) | \(-3631696/24375\) | \(-399360000\) | \([2]\) | \(6144\) | \(0.33381\) | \(\Gamma_0(N)\)-optimal |
12480.cm3 | 12480x2 | \([0, 1, 0, -2081, 35775]\) | \(15214885924/38025\) | \(2492006400\) | \([2, 2]\) | \(12288\) | \(0.68038\) | |
12480.cm2 | 12480x3 | \([0, 1, 0, -2881, 4895]\) | \(20183398562/11567205\) | \(1516136693760\) | \([2]\) | \(24576\) | \(1.0270\) | |
12480.cm1 | 12480x4 | \([0, 1, 0, -33281, 2325855]\) | \(31103978031362/195\) | \(25559040\) | \([2]\) | \(24576\) | \(1.0270\) |
Rank
sage: E.rank()
The elliptic curves in class 12480x have rank \(0\).
Complex multiplication
The elliptic curves in class 12480x do not have complex multiplication.Modular form 12480.2.a.x
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.