Properties

Label 361998.k
Number of curves $4$
Conductor $361998$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 361998.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
361998.k1 361998k3 \([1, -1, 0, -25098813, 48404327895]\) \(496930471478093017/250614\) \(881846448919254\) \([2]\) \(14450688\) \(2.6367\)  
361998.k2 361998k4 \([1, -1, 0, -1888353, 426518379]\) \(211634149400857/100188617802\) \(352538073814001033322\) \([2]\) \(14450688\) \(2.6367\)  
361998.k3 361998k2 \([1, -1, 0, -1568943, 756341145]\) \(121382959848697/86155524\) \(303159212550685764\) \([2, 2]\) \(7225344\) \(2.2902\)  
361998.k4 361998k1 \([1, -1, 0, -78363, 16715349]\) \(-15124197817/25469808\) \(-89621727993867888\) \([2]\) \(3612672\) \(1.9436\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 361998.k have rank \(1\).

Complex multiplication

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

Modular form 361998.2.a.k

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

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.