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SageMath
E = EllipticCurve("gd1")
E.isogeny_class()
Elliptic curves in class 110400gd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
110400.i4 | 110400gd1 | \([0, -1, 0, -433, -33263]\) | \(-35152/1863\) | \(-476928000000\) | \([2]\) | \(131072\) | \(0.92080\) | \(\Gamma_0(N)\)-optimal |
110400.i3 | 110400gd2 | \([0, -1, 0, -18433, -951263]\) | \(676449508/4761\) | \(4875264000000\) | \([2, 2]\) | \(262144\) | \(1.2674\) | |
110400.i2 | 110400gd3 | \([0, -1, 0, -30433, 452737]\) | \(1522096994/839523\) | \(1719343104000000\) | \([2]\) | \(524288\) | \(1.6140\) | |
110400.i1 | 110400gd4 | \([0, -1, 0, -294433, -61395263]\) | \(1378334691074/69\) | \(141312000000\) | \([2]\) | \(524288\) | \(1.6140\) |
Rank
sage: E.rank()
The elliptic curves in class 110400gd have rank \(0\).
Complex multiplication
The elliptic curves in class 110400gd do not have complex multiplication.Modular form 110400.2.a.gd
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 Cremona 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 Cremona labels.