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Properties

Label 422142z
Number of curves $4$
Conductor $422142$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("z1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

Complex multiplication

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

Modular form 422142.2.a.z

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 422142z

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
422142.z4 422142z1 \([1, 0, 1, -2025817, -528014356]\) \(6210935535799513/2781216712848\) \(411719888588111401872\) \([2]\) \(16760832\) \(2.6510\) \(\Gamma_0(N)\)-optimal
422142.z2 422142z2 \([1, 0, 1, -27428397, -55265493740]\) \(15415512004366353433/8731989540036\) \(1292647834297930352004\) \([2, 2]\) \(33521664\) \(2.9976\)  
422142.z3 422142z3 \([1, 0, 1, -22503407, -75735722176]\) \(-8513369695913641273/11811019422050982\) \(-1748454760139583323692998\) \([2]\) \(67043328\) \(3.3441\)  
422142.z1 422142z4 \([1, 0, 1, -438794667, -3537892335560]\) \(63116181515354994609913/20268303846\) \(3000436378364729094\) \([2]\) \(67043328\) \(3.3441\)