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Properties

Label 422142c
Number of curves $2$
Conductor $422142$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("c1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 422142c have rank \(0\).

Complex multiplication

The elliptic curves in class 422142c do not have complex multiplication.

Modular form 422142.2.a.c

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

Isogeny matrix

Copy content 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

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 422142c

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
422142.c1 422142c1 \([1, 1, 0, -322436, 70107024]\) \(25043174237593/94559808\) \(13998245240949312\) \([2]\) \(6285312\) \(1.9573\) \(\Gamma_0(N)\)-optimal
422142.c2 422142c2 \([1, 1, 0, -174316, 134953960]\) \(-3957057343513/50915390904\) \(-7537305156256153656\) \([2]\) \(12570624\) \(2.3039\)