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SageMath
E = EllipticCurve("gz1")
E.isogeny_class()
Elliptic curves in class 344850.gz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
344850.gz1 | 344850gz3 | \([1, 0, 0, -1878588, 989452542]\) | \(26487576322129/44531250\) | \(1232653527832031250\) | \([2]\) | \(7864320\) | \(2.3666\) | |
344850.gz2 | 344850gz2 | \([1, 0, 0, -154338, 4905792]\) | \(14688124849/8122500\) | \(224836003476562500\) | \([2, 2]\) | \(3932160\) | \(2.0200\) | |
344850.gz3 | 344850gz1 | \([1, 0, 0, -93838, -11005708]\) | \(3301293169/22800\) | \(631118606250000\) | \([2]\) | \(1966080\) | \(1.6734\) | \(\Gamma_0(N)\)-optimal |
344850.gz4 | 344850gz4 | \([1, 0, 0, 601912, 38937042]\) | \(871257511151/527800050\) | \(-14609843505907031250\) | \([2]\) | \(7864320\) | \(2.3666\) |
Rank
sage: E.rank()
The elliptic curves in class 344850.gz have rank \(1\).
Complex multiplication
The elliptic curves in class 344850.gz do not have complex multiplication.Modular form 344850.2.a.gz
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.