Properties

Label 27a
Number of curves $4$
Conductor $27$
CM \(\Q(\sqrt{-3}) \)
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 27a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
27.a3 27a1 \([0, 0, 1, 0, -7]\) \(0\) \(-19683\) \([3]\) \(1\) \(-0.49716\) \(\Gamma_0(N)\)-optimal \(-3\)
27.a1 27a2 \([0, 0, 1, -270, -1708]\) \(-12288000\) \(-177147\) \([]\) \(3\) \(0.052148\)   \(-27\)
27.a4 27a3 \([0, 0, 1, 0, 0]\) \(0\) \(-27\) \([3]\) \(3\) \(-1.0465\)   \(-3\)
27.a2 27a4 \([0, 0, 1, -30, 63]\) \(-12288000\) \(-243\) \([3]\) \(9\) \(-0.49716\)   \(-27\)

Rank

sage: E.rank()
 

The elliptic curves in class 27a have rank \(0\).

Complex multiplication

Each elliptic curve in class 27a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 27.2.a.a

sage: E.q_eigenform(10)
 
\(q - 2q^{4} - q^{7} + 5q^{13} + 4q^{16} - 7q^{19} + O(q^{20})\)  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 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.