Properties

Label 432.e
Number of curves $4$
Conductor $432$
CM \(\Q(\sqrt{-3}) \)
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 432.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
432.e1 432a4 \([0, 0, 0, -4320, 109296]\) \(-12288000\) \(-725594112\) \([]\) \(216\) \(0.74530\)   \(-27\)
432.e2 432a2 \([0, 0, 0, -480, -4048]\) \(-12288000\) \(-995328\) \([]\) \(72\) \(0.19599\)   \(-27\)
432.e3 432a1 \([0, 0, 0, 0, -16]\) \(0\) \(-110592\) \([]\) \(24\) \(-0.35332\) \(\Gamma_0(N)\)-optimal \(-3\)
432.e4 432a3 \([0, 0, 0, 0, 432]\) \(0\) \(-80621568\) \([]\) \(72\) \(0.19599\)   \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 432.e have rank \(0\).

Complex multiplication

Each elliptic curve in class 432.e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 432.2.a.e

sage: E.q_eigenform(10)
 
\(q + q^{7} + 5 q^{13} + 7 q^{19} + O(q^{20})\)  Toggle raw display

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.

\(\left(\begin{array}{rrrr} 1 & 27 & 9 & 3 \\ 27 & 1 & 3 & 9 \\ 9 & 3 & 1 & 3 \\ 3 & 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.