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SageMath
E = EllipticCurve("gt1")
E.isogeny_class()
Elliptic curves in class 133200.gt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
133200.gt1 | 133200dr4 | \([0, 0, 0, -1153650675, -3104981180750]\) | \(3639478711331685826729/2016912141902025000\) | \(94101052892580878400000000000\) | \([2]\) | \(106168320\) | \(4.2500\) | |
133200.gt2 | 133200dr2 | \([0, 0, 0, -703650675, 7141968819250]\) | \(825824067562227826729/5613755625000000\) | \(261915382440000000000000000\) | \([2, 2]\) | \(53084160\) | \(3.9034\) | |
133200.gt3 | 133200dr1 | \([0, 0, 0, -702498675, 7166652723250]\) | \(821774646379511057449/38361600000\) | \(1789798809600000000000\) | \([2]\) | \(26542080\) | \(3.5568\) | \(\Gamma_0(N)\)-optimal |
133200.gt4 | 133200dr3 | \([0, 0, 0, -272082675, 15809148963250]\) | \(-47744008200656797609/2286529541015625000\) | \(-106680322265625000000000000000\) | \([2]\) | \(106168320\) | \(4.2500\) |
Rank
sage: E.rank()
The elliptic curves in class 133200.gt have rank \(0\).
Complex multiplication
The elliptic curves in class 133200.gt do not have complex multiplication.Modular form 133200.2.a.gt
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.