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Properties

Label 421800.r
Number of curves $6$
Conductor $421800$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("r1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 421800.r have rank \(0\).

Complex multiplication

The elliptic curves in class 421800.r do not have complex multiplication.

Modular form 421800.2.a.r

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{9} + 4 q^{11} + 2 q^{13} - 2 q^{17} + q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 421800.r

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
421800.r1 421800r6 \([0, -1, 0, -249705608, -1518681688788]\) \(53809458751271244978434/234099\) \(7491168000000\) \([2]\) \(27262976\) \(3.0387\)  
421800.r2 421800r4 \([0, -1, 0, -15606608, -23725474788]\) \(26274189238602645028/54802341801\) \(876837468816000000\) \([2, 2]\) \(13631488\) \(2.6921\)  
421800.r3 421800r5 \([0, -1, 0, -15435608, -24270964788]\) \(-12709983426958940834/600633986620491\) \(-19220287571855712000000\) \([2]\) \(27262976\) \(3.0387\)  
421800.r4 421800r2 \([0, -1, 0, -986108, -361915788]\) \(26511701882112592/1170544394889\) \(4682177579556000000\) \([2, 2]\) \(6815744\) \(2.3455\)  
421800.r5 421800r1 \([0, -1, 0, -165983, 18622212]\) \(2022912739489792/574975052397\) \(143743763099250000\) \([2]\) \(3407872\) \(1.9989\) \(\Gamma_0(N)\)-optimal*
421800.r6 421800r3 \([0, -1, 0, 512392, -1365910788]\) \(929843593713212/50899738433877\) \(-814395814942032000000\) \([2]\) \(13631488\) \(2.6921\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 421800.r1.