Properties

Label 36.a
Number of curves $4$
Conductor $36$
CM \(\Q(\sqrt{-3}) \)
Rank $0$
Graph

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

Elliptic curves in class 36.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
36.a1 36a4 \([0, 0, 0, -135, -594]\) \(54000\) \(5038848\) \([2]\) \(6\) \(0.080464\)   \(-12\)
36.a2 36a2 \([0, 0, 0, -15, 22]\) \(54000\) \(6912\) \([6]\) \(2\) \(-0.46884\)   \(-12\)
36.a3 36a3 \([0, 0, 0, 0, -27]\) \(0\) \(-314928\) \([2]\) \(3\) \(-0.26611\)   \(-3\)
36.a4 36a1 \([0, 0, 0, 0, 1]\) \(0\) \(-432\) \([6]\) \(1\) \(-0.81542\) \(\Gamma_0(N)\)-optimal \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 36.a have rank \(0\).

Complex multiplication

Each elliptic curve in class 36.a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 36.2.a.a

sage: E.q_eigenform(10)
 
\(q - 4 q^{7} + 2 q^{13} + 8 q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.