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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 432a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
432.e3 | 432a1 | \([0, 0, 0, 0, -16]\) | \(0\) | \(-110592\) | \([]\) | \(24\) | \(-0.35332\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
432.e2 | 432a2 | \([0, 0, 0, -480, -4048]\) | \(-12288000\) | \(-995328\) | \([]\) | \(72\) | \(0.19599\) | \(-27\) | |
432.e4 | 432a3 | \([0, 0, 0, 0, 432]\) | \(0\) | \(-80621568\) | \([]\) | \(72\) | \(0.19599\) | \(-3\) | |
432.e1 | 432a4 | \([0, 0, 0, -4320, 109296]\) | \(-12288000\) | \(-725594112\) | \([]\) | \(216\) | \(0.74530\) | \(-27\) |
Rank
sage: E.rank()
The elliptic curves in class 432a have rank \(0\).
Complex multiplication
Each elliptic curve in class 432a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 432.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 3 & 9 \\ 3 & 1 & 9 & 27 \\ 3 & 9 & 1 & 3 \\ 9 & 27 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.