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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 76230cz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
76230.ed3 | 76230cz1 | \([1, -1, 1, -31967, -10445641]\) | \(-75526045083/943250000\) | \(-45117672657750000\) | \([2]\) | \(829440\) | \(1.8774\) | \(\Gamma_0(N)\)-optimal |
76230.ed2 | 76230cz2 | \([1, -1, 1, -939467, -349124641]\) | \(1917114236485083/7117764500\) | \(340457957875381500\) | \([2]\) | \(1658880\) | \(2.2240\) | |
76230.ed4 | 76230cz3 | \([1, -1, 1, 285658, 271139509]\) | \(73929353373/954060800\) | \(-33267752019319910400\) | \([2]\) | \(2488320\) | \(2.4267\) | |
76230.ed1 | 76230cz4 | \([1, -1, 1, -4941542, 3955270069]\) | \(382704614800227/27778076480\) | \(968611392387511266240\) | \([2]\) | \(4976640\) | \(2.7733\) |
Rank
sage: E.rank()
The elliptic curves in class 76230cz have rank \(0\).
Complex multiplication
The elliptic curves in class 76230cz do not have complex multiplication.Modular form 76230.2.a.cz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.