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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 12480bl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12480.o6 | 12480bl1 | \([0, -1, 0, 959, -4895]\) | \(371694959/249600\) | \(-65431142400\) | \([2]\) | \(12288\) | \(0.76407\) | \(\Gamma_0(N)\)-optimal |
12480.o5 | 12480bl2 | \([0, -1, 0, -4161, -36639]\) | \(30400540561/15210000\) | \(3987210240000\) | \([2, 2]\) | \(24576\) | \(1.1106\) | |
12480.o2 | 12480bl3 | \([0, -1, 0, -54081, -4818975]\) | \(66730743078481/60937500\) | \(15974400000000\) | \([2]\) | \(49152\) | \(1.4572\) | |
12480.o3 | 12480bl4 | \([0, -1, 0, -36161, 2632161]\) | \(19948814692561/231344100\) | \(60645467750400\) | \([2, 2]\) | \(49152\) | \(1.4572\) | |
12480.o1 | 12480bl5 | \([0, -1, 0, -576961, 168874081]\) | \(81025909800741361/11088090\) | \(2906676264960\) | \([2]\) | \(98304\) | \(1.8038\) | |
12480.o4 | 12480bl6 | \([0, -1, 0, -7361, 6681441]\) | \(-168288035761/73415764890\) | \(-19245502271324160\) | \([2]\) | \(98304\) | \(1.8038\) |
Rank
sage: E.rank()
The elliptic curves in class 12480bl have rank \(0\).
Complex multiplication
The elliptic curves in class 12480bl do not have complex multiplication.Modular form 12480.2.a.bl
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.