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Properties

Label 488189e
Number of curves $2$
Conductor $488189$
CM no
Rank $0$
Graph

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

Elliptic curves in class 488189e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
488189.e1 488189e1 \([1, 1, 0, -131481, -18403216]\) \(23320116793/2873\) \(30968685640217\) \([2]\) \(2525952\) \(1.6121\) \(\Gamma_0(N)\)-optimal
488189.e2 488189e2 \([1, 1, 0, -120436, -21608475]\) \(-17923019113/8254129\) \(-88973033844343441\) \([2]\) \(5051904\) \(1.9586\)  

Rank

sage: E.rank()
 

The elliptic curves in class 488189e have rank \(0\).

Complex multiplication

The elliptic curves in class 488189e do not have complex multiplication.

Modular form 488189.2.a.e

sage: E.q_eigenform(10)
 
\(q + q^{2} + 2 q^{3} - q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} - 3 q^{8} + q^{9} - 2 q^{10} + 6 q^{11} - 2 q^{12} + q^{13} + 2 q^{14} - 4 q^{15} - q^{16} + q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.