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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 12480.bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12480.bg1 | 12480cc3 | \([0, -1, 0, -6845985, -5333660703]\) | \(1082883335268084577352/251301565117746585\) | \(8234649685778320097280\) | \([2]\) | \(860160\) | \(2.9173\) | |
12480.bg2 | 12480cc2 | \([0, -1, 0, -6406585, -6238912583]\) | \(7099759044484031233216/577161945398025\) | \(2364055328350310400\) | \([2, 2]\) | \(430080\) | \(2.5707\) | |
12480.bg3 | 12480cc1 | \([0, -1, 0, -6406460, -6239168358]\) | \(454357982636417669333824/3003024375\) | \(192193560000\) | \([2]\) | \(215040\) | \(2.2241\) | \(\Gamma_0(N)\)-optimal |
12480.bg4 | 12480cc4 | \([0, -1, 0, -5969185, -7127796863]\) | \(-717825640026599866952/254764560814329735\) | \(-8348125128763956756480\) | \([2]\) | \(860160\) | \(2.9173\) |
Rank
sage: E.rank()
The elliptic curves in class 12480.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 12480.bg 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 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.