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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 6720ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6720.bw5 | 6720ca1 | \([0, 1, 0, -61, -781]\) | \(-24918016/229635\) | \(-235146240\) | \([2]\) | \(2048\) | \(0.28876\) | \(\Gamma_0(N)\)-optimal |
6720.bw4 | 6720ca2 | \([0, 1, 0, -1681, -27025]\) | \(32082281296/99225\) | \(1625702400\) | \([2, 2]\) | \(4096\) | \(0.63534\) | |
6720.bw1 | 6720ca3 | \([0, 1, 0, -26881, -1705345]\) | \(32779037733124/315\) | \(20643840\) | \([2]\) | \(8192\) | \(0.98191\) | |
6720.bw3 | 6720ca4 | \([0, 1, 0, -2401, -2401]\) | \(23366901604/13505625\) | \(885104640000\) | \([2, 2]\) | \(8192\) | \(0.98191\) | |
6720.bw2 | 6720ca5 | \([0, 1, 0, -25921, 1592255]\) | \(14695548366242/57421875\) | \(7526400000000\) | \([2]\) | \(16384\) | \(1.3285\) | |
6720.bw6 | 6720ca6 | \([0, 1, 0, 9599, -9601]\) | \(746185003198/432360075\) | \(-56670299750400\) | \([2]\) | \(16384\) | \(1.3285\) |
Rank
sage: E.rank()
The elliptic curves in class 6720ca have rank \(0\).
Complex multiplication
The elliptic curves in class 6720ca do not have complex multiplication.Modular form 6720.2.a.ca
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.