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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 6800.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6800.b1 | 6800t4 | \([0, 1, 0, -45208, 2541588]\) | \(159661140625/48275138\) | \(3089608832000000\) | \([2]\) | \(41472\) | \(1.6774\) | |
6800.b2 | 6800t3 | \([0, 1, 0, -41208, 3205588]\) | \(120920208625/19652\) | \(1257728000000\) | \([2]\) | \(20736\) | \(1.3308\) | |
6800.b3 | 6800t2 | \([0, 1, 0, -17208, -874412]\) | \(8805624625/2312\) | \(147968000000\) | \([2]\) | \(13824\) | \(1.1280\) | |
6800.b4 | 6800t1 | \([0, 1, 0, -1208, -10412]\) | \(3048625/1088\) | \(69632000000\) | \([2]\) | \(6912\) | \(0.78147\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6800.b have rank \(1\).
Complex multiplication
The elliptic curves in class 6800.b do not have complex multiplication.Modular form 6800.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.