Properties

Label 100014.e
Number of curves $2$
Conductor $100014$
CM no
Rank $1$
Graph

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

Elliptic curves in class 100014.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
100014.e1 100014f2 \([1, 0, 0, -87037, -9623389]\) \(72918170522696196433/2250945132306174\) \(2250945132306174\) \([]\) \(1213920\) \(1.7214\)  
100014.e2 100014f1 \([1, 0, 0, -11707, 482009]\) \(177444640175483953/1913455446264\) \(1913455446264\) \([3]\) \(404640\) \(1.1721\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 100014.e have rank \(1\).

Complex multiplication

The elliptic curves in class 100014.e do not have complex multiplication.

Modular form 100014.2.a.e

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.