Properties

Label 1728.n
Number of curves $4$
Conductor $1728$
CM \(\Q(\sqrt{-3}) \)
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("n1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1728.n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
1728.n1 1728a4 \([0, 0, 0, -1080, -13662]\) \(-12288000\) \(-11337408\) \([]\) \(432\) \(0.39872\)   \(-27\)
1728.n2 1728a3 \([0, 0, 0, -120, 506]\) \(-12288000\) \(-15552\) \([]\) \(144\) \(-0.15058\)   \(-27\)
1728.n3 1728a2 \([0, 0, 0, 0, -54]\) \(0\) \(-1259712\) \([]\) \(144\) \(-0.15058\)   \(-3\)
1728.n4 1728a1 \([0, 0, 0, 0, 2]\) \(0\) \(-1728\) \([]\) \(48\) \(-0.69989\) \(\Gamma_0(N)\)-optimal \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 1728.n have rank \(1\).

Complex multiplication

Each elliptic curve in class 1728.n has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 1728.2.a.n

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.