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Properties

Label 231231.ce
Number of curves $2$
Conductor $231231$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ce1")
 
E.isogeny_class()
 

Elliptic curves in class 231231.ce

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
231231.ce1 231231ce2 \([1, 0, 1, -77201, 6203789]\) \(244140625/61347\) \(12786087751319883\) \([2]\) \(1474560\) \(1.8005\)  
231231.ce2 231231ce1 \([1, 0, 1, 11734, 618671]\) \(857375/1287\) \(-268239603174543\) \([2]\) \(737280\) \(1.4540\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 231231.ce have rank \(1\).

Complex multiplication

The elliptic curves in class 231231.ce do not have complex multiplication.

Modular form 231231.2.a.ce

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} + q^{6} - 3 q^{8} + q^{9} - q^{12} + q^{13} - q^{16} - 4 q^{17} + q^{18} - 8 q^{19} + O(q^{20})\) Copy content 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.