Properties

Label 441b
Number of curves $2$
Conductor $441$
CM \(\Q(\sqrt{-3}) \)
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 441b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
441.d2 441b1 \([0, 0, 1, 0, 12]\) \(0\) \(-64827\) \([3]\) \(24\) \(-0.39783\) \(\Gamma_0(N)\)-optimal \(-3\)
441.d1 441b2 \([0, 0, 1, 0, -331]\) \(0\) \(-47258883\) \([]\) \(72\) \(0.15148\)   \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 441b have rank \(1\).

Complex multiplication

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

Modular form 441.2.a.b

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 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.