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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 16800.ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16800.ca1 | 16800by2 | \([0, 1, 0, -2100008, -1172031012]\) | \(128025588102048008/7875\) | \(63000000000\) | \([2]\) | \(147456\) | \(1.9807\) | |
16800.ca2 | 16800by3 | \([0, 1, 0, -147008, -13681512]\) | \(43919722445768/15380859375\) | \(123046875000000000\) | \([2]\) | \(147456\) | \(1.9807\) | |
16800.ca3 | 16800by1 | \([0, 1, 0, -131258, -18343512]\) | \(250094631024064/62015625\) | \(62015625000000\) | \([2, 2]\) | \(73728\) | \(1.6341\) | \(\Gamma_0(N)\)-optimal |
16800.ca4 | 16800by4 | \([0, 1, 0, -115633, -22859137]\) | \(-2671731885376/1969120125\) | \(-126023688000000000\) | \([4]\) | \(147456\) | \(1.9807\) |
Rank
sage: E.rank()
The elliptic curves in class 16800.ca have rank \(0\).
Complex multiplication
The elliptic curves in class 16800.ca do not have complex multiplication.Modular form 16800.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.