Properties

Label 441.c
Number of curves $4$
Conductor $441$
CM \(\Q(\sqrt{-7}) \)
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 441.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
441.c1 441d4 \([1, -1, 1, -16400, -804212]\) \(16581375\) \(29417779503\) \([2]\) \(448\) \(1.0697\)   \(-28\)
441.c2 441d3 \([1, -1, 1, -965, -13940]\) \(-3375\) \(-29417779503\) \([2]\) \(224\) \(0.72312\)   \(-7\)
441.c3 441d2 \([1, -1, 1, -335, 2440]\) \(16581375\) \(250047\) \([2]\) \(64\) \(0.096741\)   \(-28\)
441.c4 441d1 \([1, -1, 1, -20, 46]\) \(-3375\) \(-250047\) \([2]\) \(32\) \(-0.24983\) \(\Gamma_0(N)\)-optimal \(-7\)

Rank

sage: E.rank()
 

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

Complex multiplication

Each elliptic curve in class 441.c has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).

Modular form 441.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + 3q^{8} - 4q^{11} - q^{16} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.