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Properties

Label 100010e
Number of curves $2$
Conductor $100010$
CM no
Rank $1$
Graph

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

Elliptic curves in class 100010e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
100010.f1 100010e1 \([1, -1, 1, -17216328052, -869056815249449]\) \(564345804012377540082892046274202641/313563965869260800000000000000\) \(313563965869260800000000000000\) \([2]\) \(222813696\) \(4.6074\) \(\Gamma_0(N)\)-optimal
100010.f2 100010e2 \([1, -1, 1, -14154486132, -1187958673848361]\) \(-313621648911503266976083898753158161/428167812500000000000000000000000\) \(-428167812500000000000000000000000\) \([2]\) \(445627392\) \(4.9540\)  

Rank

sage: E.rank()
 

The elliptic curves in class 100010e have rank \(1\).

Complex multiplication

The elliptic curves in class 100010e do not have complex multiplication.

Modular form 100010.2.a.e

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