Properties

Label 256.a
Number of curves $2$
Conductor $256$
CM \(\Q(\sqrt{-2}) \)
Rank $1$
Graph

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

Elliptic curves in class 256.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
256.a1 256a2 \([0, 1, 0, -13, -21]\) \(8000\) \(32768\) \([2]\) \(16\) \(-0.40397\)   \(-8\)
256.a2 256a1 \([0, 1, 0, -3, 1]\) \(8000\) \(512\) \([2]\) \(8\) \(-0.75055\) \(\Gamma_0(N)\)-optimal \(-8\)

Rank

sage: E.rank()
 

The elliptic curves in class 256.a have rank \(1\).

Complex multiplication

Each elliptic curve in class 256.a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-2}) \).

Modular form 256.2.a.a

sage: E.q_eigenform(10)
 
\(q - 2q^{3} + q^{9} - 6q^{11} - 6q^{17} - 2q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.