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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1728a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
1728.n4 | 1728a1 | \([0, 0, 0, 0, 2]\) | \(0\) | \(-1728\) | \([]\) | \(48\) | \(-0.69989\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
1728.n3 | 1728a2 | \([0, 0, 0, 0, -54]\) | \(0\) | \(-1259712\) | \([]\) | \(144\) | \(-0.15058\) | \(-3\) | |
1728.n2 | 1728a3 | \([0, 0, 0, -120, 506]\) | \(-12288000\) | \(-15552\) | \([]\) | \(144\) | \(-0.15058\) | \(-27\) | |
1728.n1 | 1728a4 | \([0, 0, 0, -1080, -13662]\) | \(-12288000\) | \(-11337408\) | \([]\) | \(432\) | \(0.39872\) | \(-27\) |
Rank
sage: E.rank()
The elliptic curves in class 1728a have rank \(1\).
Complex multiplication
Each elliptic curve in class 1728a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 1728.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 3 & 9 \\ 3 & 1 & 9 & 3 \\ 3 & 9 & 1 & 27 \\ 9 & 3 & 27 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.