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Properties

Label 487872en
Number of curves $6$
Conductor $487872$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("en1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 487872en have rank \(0\).

Complex multiplication

The elliptic curves in class 487872en do not have complex multiplication.

Modular form 487872.2.a.en

Copy content sage:E.q_eigenform(10)
 
\(q - 2 q^{5} + q^{7} - 2 q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 487872en

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
487872.en4 487872en1 \([0, 0, 0, -9057576, 10492187816]\) \(62140690757632/6237\) \(8248202979996672\) \([2]\) \(11796480\) \(2.4854\) \(\Gamma_0(N)\)-optimal
487872.en3 487872en2 \([0, 0, 0, -9079356, 10439192720]\) \(3911877700432/38900169\) \(823104671779827892224\) \([2, 2]\) \(23592960\) \(2.8320\)  
487872.en5 487872en3 \([0, 0, 0, -2371116, 25521999536]\) \(-17418812548/3314597517\) \(-280539727353114327318528\) \([2]\) \(47185920\) \(3.1786\)  
487872.en2 487872en4 \([0, 0, 0, -16136076, -8035300240]\) \(5489767279588/2847396321\) \(240996918468276522123264\) \([2, 2]\) \(47185920\) \(3.1786\)  
487872.en6 487872en5 \([0, 0, 0, 60703764, -62468642896]\) \(146142660369886/94532266521\) \(-16001976794984292790960128\) \([2]\) \(94371840\) \(3.5251\)  
487872.en1 487872en6 \([0, 0, 0, -205883436, -1135969507024]\) \(5701568801608514/6277868289\) \(1062687972896844682493952\) \([2]\) \(94371840\) \(3.5251\)