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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 2112k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2112.z3 | 2112k1 | \([0, 1, 0, -132, -630]\) | \(4004529472/99\) | \(6336\) | \([2]\) | \(256\) | \(-0.16135\) | \(\Gamma_0(N)\)-optimal |
2112.z2 | 2112k2 | \([0, 1, 0, -137, -585]\) | \(69934528/9801\) | \(40144896\) | \([2, 2]\) | \(512\) | \(0.18522\) | |
2112.z1 | 2112k3 | \([0, 1, 0, -577, 4607]\) | \(649461896/72171\) | \(2364899328\) | \([4]\) | \(1024\) | \(0.53179\) | |
2112.z4 | 2112k4 | \([0, 1, 0, 223, -2817]\) | \(37259704/131769\) | \(-4317806592\) | \([2]\) | \(1024\) | \(0.53179\) |
Rank
sage: E.rank()
The elliptic curves in class 2112k have rank \(0\).
Complex multiplication
The elliptic curves in class 2112k do not have complex multiplication.Modular form 2112.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.