Properties

Label 13860.k
Number of curves $4$
Conductor $13860$
CM no
Rank $0$
Graph

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

Elliptic curves in class 13860.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13860.k1 13860c3 \([0, 0, 0, -9288, 27837]\) \(281370820608/161767375\) \(50945075874000\) \([2]\) \(31104\) \(1.3198\)  
13860.k2 13860c1 \([0, 0, 0, -6648, 208633]\) \(75216478666752/326095\) \(140873040\) \([6]\) \(10368\) \(0.77046\) \(\Gamma_0(N)\)-optimal
13860.k3 13860c2 \([0, 0, 0, -6543, 215542]\) \(-4481782160112/310023175\) \(-2142880185600\) \([6]\) \(20736\) \(1.1170\)  
13860.k4 13860c4 \([0, 0, 0, 37017, 222318]\) \(1113258734352/648484375\) \(-3267614196000000\) \([2]\) \(62208\) \(1.6663\)  

Rank

sage: E.rank()
 

The elliptic curves in class 13860.k have rank \(0\).

Complex multiplication

The elliptic curves in class 13860.k do not have complex multiplication.

Modular form 13860.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{5} + q^{7} + q^{11} + 2q^{13} - 6q^{17} - 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.