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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 100.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100.a1 | 100a3 | \([0, -1, 0, -1033, -12438]\) | \(488095744/125\) | \(31250000\) | \([2]\) | \(36\) | \(0.42408\) | |
100.a2 | 100a4 | \([0, -1, 0, -908, -15688]\) | \(-20720464/15625\) | \(-62500000000\) | \([2]\) | \(72\) | \(0.77065\) | |
100.a3 | 100a1 | \([0, -1, 0, -33, 62]\) | \(16384/5\) | \(1250000\) | \([2]\) | \(12\) | \(-0.12523\) | \(\Gamma_0(N)\)-optimal |
100.a4 | 100a2 | \([0, -1, 0, 92, 312]\) | \(21296/25\) | \(-100000000\) | \([2]\) | \(24\) | \(0.22134\) |
Rank
sage: E.rank()
The elliptic curves in class 100.a have rank \(0\).
Complex multiplication
The elliptic curves in class 100.a do not have complex multiplication.Modular form 100.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.