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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1200j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1200.e7 | 1200j1 | \([0, -1, 0, -8, -1488]\) | \(-1/15\) | \(-960000000\) | \([2]\) | \(384\) | \(0.40244\) | \(\Gamma_0(N)\)-optimal |
1200.e6 | 1200j2 | \([0, -1, 0, -2008, -33488]\) | \(13997521/225\) | \(14400000000\) | \([2, 2]\) | \(768\) | \(0.74901\) | |
1200.e4 | 1200j3 | \([0, -1, 0, -32008, -2193488]\) | \(56667352321/15\) | \(960000000\) | \([2]\) | \(1536\) | \(1.0956\) | |
1200.e5 | 1200j4 | \([0, -1, 0, -4008, 46512]\) | \(111284641/50625\) | \(3240000000000\) | \([2, 2]\) | \(1536\) | \(1.0956\) | |
1200.e2 | 1200j5 | \([0, -1, 0, -54008, 4846512]\) | \(272223782641/164025\) | \(10497600000000\) | \([2, 2]\) | \(3072\) | \(1.4422\) | |
1200.e8 | 1200j6 | \([0, -1, 0, 13992, 334512]\) | \(4733169839/3515625\) | \(-225000000000000\) | \([2]\) | \(3072\) | \(1.4422\) | |
1200.e1 | 1200j7 | \([0, -1, 0, -864008, 309406512]\) | \(1114544804970241/405\) | \(25920000000\) | \([4]\) | \(6144\) | \(1.7887\) | |
1200.e3 | 1200j8 | \([0, -1, 0, -44008, 6686512]\) | \(-147281603041/215233605\) | \(-13774950720000000\) | \([2]\) | \(6144\) | \(1.7887\) |
Rank
sage: E.rank()
The elliptic curves in class 1200j have rank \(0\).
Complex multiplication
The elliptic curves in class 1200j do not have complex multiplication.Modular form 1200.2.a.j
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.