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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 2448n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2448.p5 | 2448n1 | \([0, 0, 0, -4899, 122402]\) | \(4354703137/352512\) | \(1052595191808\) | \([2]\) | \(3072\) | \(1.0498\) | \(\Gamma_0(N)\)-optimal |
| 2448.p4 | 2448n2 | \([0, 0, 0, -16419, -667870]\) | \(163936758817/30338064\) | \(90588973694976\) | \([2, 2]\) | \(6144\) | \(1.3964\) | |
| 2448.p2 | 2448n3 | \([0, 0, 0, -249699, -48023710]\) | \(576615941610337/27060804\) | \(80803127771136\) | \([2, 2]\) | \(12288\) | \(1.7430\) | |
| 2448.p6 | 2448n4 | \([0, 0, 0, 32541, -3889438]\) | \(1276229915423/2927177028\) | \(-8740503770775552\) | \([2]\) | \(12288\) | \(1.7430\) | |
| 2448.p1 | 2448n5 | \([0, 0, 0, -3995139, -3073590142]\) | \(2361739090258884097/5202\) | \(15533088768\) | \([2]\) | \(24576\) | \(2.0895\) | |
| 2448.p3 | 2448n6 | \([0, 0, 0, -236739, -53231038]\) | \(-491411892194497/125563633938\) | \(-374931001920724992\) | \([2]\) | \(24576\) | \(2.0895\) |
Rank
sage: E.rank()
The elliptic curves in class 2448n have rank \(1\).
Complex multiplication
The elliptic curves in class 2448n do not have complex multiplication.Modular form 2448.2.a.n
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
