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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 1200p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1200.k8 | 1200p1 | \([0, 1, 0, 592, -16812]\) | \(357911/2160\) | \(-138240000000\) | \([2]\) | \(1152\) | \(0.81958\) | \(\Gamma_0(N)\)-optimal |
1200.k6 | 1200p2 | \([0, 1, 0, -7408, -224812]\) | \(702595369/72900\) | \(4665600000000\) | \([2, 2]\) | \(2304\) | \(1.1662\) | |
1200.k7 | 1200p3 | \([0, 1, 0, -5408, 499188]\) | \(-273359449/1536000\) | \(-98304000000000\) | \([2]\) | \(3456\) | \(1.3689\) | |
1200.k4 | 1200p4 | \([0, 1, 0, -115408, -15128812]\) | \(2656166199049/33750\) | \(2160000000000\) | \([2]\) | \(4608\) | \(1.5127\) | |
1200.k5 | 1200p5 | \([0, 1, 0, -27408, 1495188]\) | \(35578826569/5314410\) | \(340122240000000\) | \([4]\) | \(4608\) | \(1.5127\) | |
1200.k3 | 1200p6 | \([0, 1, 0, -133408, 18675188]\) | \(4102915888729/9000000\) | \(576000000000000\) | \([2, 2]\) | \(6912\) | \(1.7155\) | |
1200.k2 | 1200p7 | \([0, 1, 0, -181408, 3987188]\) | \(10316097499609/5859375000\) | \(375000000000000000\) | \([2]\) | \(13824\) | \(2.0620\) | |
1200.k1 | 1200p8 | \([0, 1, 0, -2133408, 1198675188]\) | \(16778985534208729/81000\) | \(5184000000000\) | \([4]\) | \(13824\) | \(2.0620\) |
Rank
sage: E.rank()
The elliptic curves in class 1200p have rank \(1\).
Complex multiplication
The elliptic curves in class 1200p do not have complex multiplication.Modular form 1200.2.a.p
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.