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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 12480bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12480.cs4 | 12480bg1 | \([0, 1, 0, 335, 1775]\) | \(253012016/219375\) | \(-3594240000\) | \([2]\) | \(6144\) | \(0.52108\) | \(\Gamma_0(N)\)-optimal |
12480.cs3 | 12480bg2 | \([0, 1, 0, -1665, 14175]\) | \(7793764996/3080025\) | \(201852518400\) | \([2, 2]\) | \(12288\) | \(0.86766\) | |
12480.cs2 | 12480bg3 | \([0, 1, 0, -12065, -503745]\) | \(1481943889298/34543665\) | \(4527707258880\) | \([2]\) | \(24576\) | \(1.2142\) | |
12480.cs1 | 12480bg4 | \([0, 1, 0, -23265, 1357695]\) | \(10625310339698/3855735\) | \(505378897920\) | \([2]\) | \(24576\) | \(1.2142\) |
Rank
sage: E.rank()
The elliptic curves in class 12480bg have rank \(1\).
Complex multiplication
The elliptic curves in class 12480bg do not have complex multiplication.Modular form 12480.2.a.bg
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.