Properties

Label 32.a
Number of curves $4$
Conductor $32$
CM \(\Q(\sqrt{-1}) \)
Rank $0$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 32.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
32.a1 32a3 \([0, 0, 0, -11, -14]\) \(287496\) \(512\) \([2]\) \(4\) \(-0.61739\)   \(-16\)
32.a2 32a4 \([0, 0, 0, -11, 14]\) \(287496\) \(512\) \([4]\) \(4\) \(-0.61739\)   \(-16\)
32.a3 32a2 \([0, 0, 0, -1, 0]\) \(1728\) \(64\) \([2, 2]\) \(2\) \(-0.96396\)   \(-4\)
32.a4 32a1 \([0, 0, 0, 4, 0]\) \(1728\) \(-4096\) \([4]\) \(1\) \(-0.61739\) \(\Gamma_0(N)\)-optimal \(-4\)

Rank

sage: E.rank()
 

The elliptic curves in class 32.a have rank \(0\).

Complex multiplication

Each elliptic curve in class 32.a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 32.2.a.a

sage: E.q_eigenform(10)
 
\(q - 2q^{5} - 3q^{9} + 6q^{13} + 2q^{17} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.