Properties

Label 1764.e
Number of curves $4$
Conductor $1764$
CM \(\Q(\sqrt{-3}) \)
Rank $1$
Graph

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

Elliptic curves in class 1764.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
1764.e1 1764b4 \([0, 0, 0, -6615, 203742]\) \(54000\) \(592815428352\) \([2]\) \(1728\) \(1.0534\)   \(-12\)
1764.e2 1764b2 \([0, 0, 0, -735, -7546]\) \(54000\) \(813189888\) \([2]\) \(576\) \(0.50411\)   \(-12\)
1764.e3 1764b1 \([0, 0, 0, 0, -343]\) \(0\) \(-50824368\) \([2]\) \(288\) \(0.15754\) \(\Gamma_0(N)\)-optimal \(-3\)
1764.e4 1764b3 \([0, 0, 0, 0, 9261]\) \(0\) \(-37050964272\) \([2]\) \(864\) \(0.70685\)   \(-3\)

Rank

sage: E.rank()
 

The elliptic curves in class 1764.e have rank \(1\).

Complex multiplication

Each elliptic curve in class 1764.e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 1764.2.a.e

sage: E.q_eigenform(10)
 
\(q - 2q^{13} - 8q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.