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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 6720.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6720.b1 | 6720e4 | \([0, -1, 0, -26881, 1705345]\) | \(32779037733124/315\) | \(20643840\) | \([2]\) | \(8192\) | \(0.98191\) | |
| 6720.b2 | 6720e5 | \([0, -1, 0, -25921, -1592255]\) | \(14695548366242/57421875\) | \(7526400000000\) | \([2]\) | \(16384\) | \(1.3285\) | |
| 6720.b3 | 6720e3 | \([0, -1, 0, -2401, 2401]\) | \(23366901604/13505625\) | \(885104640000\) | \([2, 2]\) | \(8192\) | \(0.98191\) | |
| 6720.b4 | 6720e2 | \([0, -1, 0, -1681, 27025]\) | \(32082281296/99225\) | \(1625702400\) | \([2, 2]\) | \(4096\) | \(0.63534\) | |
| 6720.b5 | 6720e1 | \([0, -1, 0, -61, 781]\) | \(-24918016/229635\) | \(-235146240\) | \([2]\) | \(2048\) | \(0.28876\) | \(\Gamma_0(N)\)-optimal |
| 6720.b6 | 6720e6 | \([0, -1, 0, 9599, 9601]\) | \(746185003198/432360075\) | \(-56670299750400\) | \([2]\) | \(16384\) | \(1.3285\) |
Rank
sage: E.rank()
The elliptic curves in class 6720.b have rank \(1\).
Complex multiplication
The elliptic curves in class 6720.b do not have complex multiplication.Modular form 6720.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
