Properties

Label 243a
Number of curves $2$
Conductor $243$
CM \(\Q(\sqrt{-3}) \)
Rank $1$
Graph

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

Elliptic curves in class 243a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
243.a1 243a1 \([0, 0, 1, 0, -1]\) \(0\) \(-243\) \([]\) \(6\) \(-0.86336\) \(\Gamma_0(N)\)-optimal \(-3\)
243.a2 243a2 \([0, 0, 1, 0, 20]\) \(0\) \(-177147\) \([3]\) \(18\) \(-0.31406\)   \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 243a have rank \(1\).

Complex multiplication

Each elliptic curve in class 243a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 243.2.a.a

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.