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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 99.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
99.b1 | 99b3 | \([1, -1, 1, -1319, -18084]\) | \(347873904937/395307\) | \(288178803\) | \([2]\) | \(48\) | \(0.53667\) | |
99.b2 | 99b2 | \([1, -1, 1, -104, -102]\) | \(169112377/88209\) | \(64304361\) | \([2, 2]\) | \(24\) | \(0.19010\) | |
99.b3 | 99b1 | \([1, -1, 1, -59, 186]\) | \(30664297/297\) | \(216513\) | \([4]\) | \(12\) | \(-0.15647\) | \(\Gamma_0(N)\)-optimal |
99.b4 | 99b4 | \([1, -1, 1, 391, -1092]\) | \(9090072503/5845851\) | \(-4261625379\) | \([2]\) | \(48\) | \(0.53667\) |
Rank
sage: E.rank()
The elliptic curves in class 99.b have rank \(0\).
Complex multiplication
The elliptic curves in class 99.b do not have complex multiplication.Modular form 99.2.a.b
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.