Properties

Label 441.e
Number of curves $2$
Conductor $441$
CM \(\Q(\sqrt{-3}) \)
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 441.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
441.e1 441a1 \([0, 0, 1, 0, -4202]\) \(0\) \(-7626831723\) \([]\) \(168\) \(0.57513\) \(\Gamma_0(N)\)-optimal \(-3\)
441.e2 441a2 \([0, 0, 1, 0, 113447]\) \(0\) \(-5559960326067\) \([]\) \(504\) \(1.1244\)   \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 441.e have rank \(0\).

Complex multiplication

Each elliptic curve in class 441.e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 441.2.a.e

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