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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 25200.l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25200.l1 | 25200bf4 | \([0, 0, 0, -67275, 6716250]\) | \(1443468546/7\) | \(163296000000\) | \([2]\) | \(65536\) | \(1.3520\) | |
25200.l2 | 25200bf3 | \([0, 0, 0, -13275, -465750]\) | \(11090466/2401\) | \(56010528000000\) | \([2]\) | \(65536\) | \(1.3520\) | |
25200.l3 | 25200bf2 | \([0, 0, 0, -4275, 101250]\) | \(740772/49\) | \(571536000000\) | \([2, 2]\) | \(32768\) | \(1.0054\) | |
25200.l4 | 25200bf1 | \([0, 0, 0, 225, 6750]\) | \(432/7\) | \(-20412000000\) | \([2]\) | \(16384\) | \(0.65884\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25200.l have rank \(2\).
Complex multiplication
The elliptic curves in class 25200.l do not have complex multiplication.Modular form 25200.2.a.l
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.