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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 8400.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8400.f1 | 8400a3 | \([0, -1, 0, -56008, 5120512]\) | \(1214399773444/105\) | \(1680000000\) | \([4]\) | \(12288\) | \(1.2107\) | |
8400.f2 | 8400a2 | \([0, -1, 0, -3508, 80512]\) | \(1193895376/11025\) | \(44100000000\) | \([2, 2]\) | \(6144\) | \(0.86414\) | |
8400.f3 | 8400a4 | \([0, -1, 0, -1008, 190512]\) | \(-7086244/972405\) | \(-15558480000000\) | \([2]\) | \(12288\) | \(1.2107\) | |
8400.f4 | 8400a1 | \([0, -1, 0, -383, -738]\) | \(24918016/13125\) | \(3281250000\) | \([2]\) | \(3072\) | \(0.51757\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8400.f have rank \(1\).
Complex multiplication
The elliptic curves in class 8400.f do not have complex multiplication.Modular form 8400.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.