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Properties

Label 180999.d
Number of curves $2$
Conductor $180999$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 180999.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
180999.d1 180999d1 \([1, -1, 1, -692087, -221319250]\) \(10418796526321/6390657\) \(22487084447440977\) \([2]\) \(3010560\) \(2.0804\) \(\Gamma_0(N)\)-optimal
180999.d2 180999d2 \([1, -1, 1, -562802, -306647350]\) \(-5602762882081/8312741073\) \(-29250405787427195553\) \([2]\) \(6021120\) \(2.4270\)  

Rank

sage: E.rank()
 

The elliptic curves in class 180999.d have rank \(2\).

Complex multiplication

The elliptic curves in class 180999.d do not have complex multiplication.

Modular form 180999.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - 4 q^{5} - q^{7} + 3 q^{8} + 4 q^{10} - 4 q^{11} + q^{14} - q^{16} - q^{17} + O(q^{20})\) 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.