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Properties

Label 324870.s
Number of curves $2$
Conductor $324870$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("s1")
 
E.isogeny_class()
 

Elliptic curves in class 324870.s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
324870.s1 324870s2 \([1, 1, 0, -133911117, 594953824221]\) \(5419728549194533398573442729/15116050674292359168000\) \(740686483040325599232000\) \([]\) \(81959040\) \(3.4526\)  
324870.s2 324870s1 \([1, 1, 0, -7977582, -7984739916]\) \(1145886354864987091635769/101502227356009758720\) \(4973609140444478177280\) \([]\) \(27319680\) \(2.9033\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 324870.s have rank \(1\).

Complex multiplication

The elliptic curves in class 324870.s do not have complex multiplication.

Modular form 324870.2.a.s

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