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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 9009.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9009.k1 | 9009c4 | \([1, -1, 0, -89241, 10283462]\) | \(107818231938348177/4463459\) | \(3253861611\) | \([2]\) | \(19456\) | \(1.3115\) | |
9009.k2 | 9009c3 | \([1, -1, 0, -9051, -59860]\) | \(112489728522417/62811265517\) | \(45789412561893\) | \([2]\) | \(19456\) | \(1.3115\) | |
9009.k3 | 9009c2 | \([1, -1, 0, -5586, 161207]\) | \(26444947540257/169338169\) | \(123447525201\) | \([2, 2]\) | \(9728\) | \(0.96496\) | |
9009.k4 | 9009c1 | \([1, -1, 0, -141, 5480]\) | \(-426957777/17320303\) | \(-12626500887\) | \([2]\) | \(4864\) | \(0.61839\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 9009.k have rank \(0\).
Complex multiplication
The elliptic curves in class 9009.k do not have complex multiplication.Modular form 9009.2.a.k
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.