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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 198.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
198.d1 | 198c4 | \([1, -1, 1, -1325, 4969]\) | \(13060888875/7086244\) | \(139478540652\) | \([2]\) | \(192\) | \(0.82936\) | |
198.d2 | 198c2 | \([1, -1, 1, -1025, 12881]\) | \(4406910829875/7744\) | \(209088\) | \([6]\) | \(64\) | \(0.28006\) | |
198.d3 | 198c3 | \([1, -1, 1, -785, -8207]\) | \(2714704875/21296\) | \(419169168\) | \([2]\) | \(96\) | \(0.48279\) | |
198.d4 | 198c1 | \([1, -1, 1, -65, 209]\) | \(1108717875/45056\) | \(1216512\) | \([6]\) | \(32\) | \(-0.066515\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 198.d have rank \(0\).
Complex multiplication
The elliptic curves in class 198.d do not have complex multiplication.Modular form 198.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.