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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 18810k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18810.j4 | 18810k1 | \([1, -1, 0, 396, -3632]\) | \(9407293631/12840960\) | \(-9361059840\) | \([2]\) | \(12288\) | \(0.59904\) | \(\Gamma_0(N)\)-optimal |
18810.j3 | 18810k2 | \([1, -1, 0, -2484, -34160]\) | \(2325676477249/629006400\) | \(458545665600\) | \([2, 2]\) | \(24576\) | \(0.94562\) | |
18810.j1 | 18810k3 | \([1, -1, 0, -36684, -2694920]\) | \(7489156350944449/901299960\) | \(657047670840\) | \([2]\) | \(49152\) | \(1.2922\) | |
18810.j2 | 18810k4 | \([1, -1, 0, -14364, 638248]\) | \(449613538734529/21502965000\) | \(15675661485000\) | \([2]\) | \(49152\) | \(1.2922\) |
Rank
sage: E.rank()
The elliptic curves in class 18810k have rank \(1\).
Complex multiplication
The elliptic curves in class 18810k do not have complex multiplication.Modular form 18810.2.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.