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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 12480cj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12480.ca6 | 12480cj1 | \([0, 1, 0, -7041, -229761]\) | \(147281603041/5265\) | \(1380188160\) | \([2]\) | \(12288\) | \(0.84284\) | \(\Gamma_0(N)\)-optimal |
12480.ca5 | 12480cj2 | \([0, 1, 0, -7361, -208065]\) | \(168288035761/27720225\) | \(7266690662400\) | \([2, 2]\) | \(24576\) | \(1.1894\) | |
12480.ca4 | 12480cj3 | \([0, 1, 0, -33281, 2129919]\) | \(15551989015681/1445900625\) | \(379034173440000\) | \([2, 2]\) | \(49152\) | \(1.5360\) | |
12480.ca7 | 12480cj4 | \([0, 1, 0, 13439, -1152385]\) | \(1023887723039/2798036865\) | \(-733488575938560\) | \([2]\) | \(49152\) | \(1.5360\) | |
12480.ca2 | 12480cj5 | \([0, 1, 0, -520001, 144154815]\) | \(59319456301170001/594140625\) | \(155750400000000\) | \([2, 2]\) | \(98304\) | \(1.8826\) | |
12480.ca8 | 12480cj6 | \([0, 1, 0, 38719, 10150719]\) | \(24487529386319/183539412225\) | \(-48113755678310400\) | \([2]\) | \(98304\) | \(1.8826\) | |
12480.ca1 | 12480cj7 | \([0, 1, 0, -8320001, 9234274815]\) | \(242970740812818720001/24375\) | \(6389760000\) | \([2]\) | \(196608\) | \(2.2291\) | |
12480.ca3 | 12480cj8 | \([0, 1, 0, -507521, 151415679]\) | \(-55150149867714721/5950927734375\) | \(-1560000000000000000\) | \([2]\) | \(196608\) | \(2.2291\) |
Rank
sage: E.rank()
The elliptic curves in class 12480cj have rank \(1\).
Complex multiplication
The elliptic curves in class 12480cj do not have complex multiplication.Modular form 12480.2.a.cj
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.