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SageMath
E = EllipticCurve("ez1")
E.isogeny_class()
Elliptic curves in class 163170.ez
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
163170.ez1 | 163170b4 | \([1, -1, 1, -4624997, -1707755619]\) | \(127568139540190201/59114336463360\) | \(5070007333951245826560\) | \([2]\) | \(13934592\) | \(2.8594\) | |
163170.ez2 | 163170b2 | \([1, -1, 1, -2342822, 1380761421]\) | \(16581570075765001/998001000\) | \(85594674524121000\) | \([2]\) | \(4644864\) | \(2.3101\) | |
163170.ez3 | 163170b1 | \([1, -1, 1, -137822, 24245421]\) | \(-3375675045001/999000000\) | \(-85680354879000000\) | \([2]\) | \(2322432\) | \(1.9635\) | \(\Gamma_0(N)\)-optimal |
163170.ez4 | 163170b3 | \([1, -1, 1, 1019803, -201722979]\) | \(1367594037332999/995878502400\) | \(-85412636138137190400\) | \([2]\) | \(6967296\) | \(2.5128\) |
Rank
sage: E.rank()
The elliptic curves in class 163170.ez have rank \(2\).
Complex multiplication
The elliptic curves in class 163170.ez do not have complex multiplication.Modular form 163170.2.a.ez
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.