Properties

Label 361998d
Number of curves $2$
Conductor $361998$
CM no
Rank $2$
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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 361998d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
361998.d1 361998d1 \([1, -1, 0, -307098, 65598196]\) \(-1999852137736669/636095936\) \(-1018779520344768\) \([]\) \(3024000\) \(1.8549\) \(\Gamma_0(N)\)-optimal
361998.d2 361998d2 \([1, -1, 0, 1950417, -195921491]\) \(512328390183400211/306788440211456\) \(-491356354092390678528\) \([]\) \(15120000\) \(2.6596\)  

Rank

sage: E.rank()
 

The elliptic curves in class 361998d have rank \(2\).

Complex multiplication

The elliptic curves in class 361998d do not have complex multiplication.

Modular form 361998.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{5} - q^{7} - q^{8} + 2 q^{10} - 5 q^{11} + q^{14} + q^{16} + q^{17} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.