Properties

Label 25410.ct
Number of curves $8$
Conductor $25410$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ct1")
 
E.isogeny_class()
 

Elliptic curves in class 25410.ct

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
25410.ct1 25410ct8 \([1, 0, 0, -42499135, 106635995225]\) \(4791901410190533590281/41160000\) \(72917450760000\) \([2]\) \(1658880\) \(2.7012\)  
25410.ct2 25410ct6 \([1, 0, 0, -2656255, 1665943577]\) \(1169975873419524361/108425318400\) \(192082065490022400\) \([2, 2]\) \(829440\) \(2.3546\)  
25410.ct3 25410ct7 \([1, 0, 0, -2462655, 1919133657]\) \(-932348627918877961/358766164249920\) \(-635576144704752525120\) \([2]\) \(1658880\) \(2.7012\)  
25410.ct4 25410ct5 \([1, 0, 0, -527260, 144727100]\) \(9150443179640281/184570312500\) \(326977567382812500\) \([2]\) \(552960\) \(2.1519\)  
25410.ct5 25410ct3 \([1, 0, 0, -178175, 21985305]\) \(353108405631241/86318776320\) \(152918977696235520\) \([2]\) \(414720\) \(2.0081\)  
25410.ct6 25410ct2 \([1, 0, 0, -69880, -3738448]\) \(21302308926361/8930250000\) \(15820482620250000\) \([2, 2]\) \(276480\) \(1.8053\)  
25410.ct7 25410ct1 \([1, 0, 0, -60200, -5688000]\) \(13619385906841/6048000\) \(10714400928000\) \([2]\) \(138240\) \(1.4588\) \(\Gamma_0(N)\)-optimal
25410.ct8 25410ct4 \([1, 0, 0, 232620, -27393948]\) \(785793873833639/637994920500\) \(-1130246919355900500\) \([2]\) \(552960\) \(2.1519\)  

Rank

sage: E.rank()
 

The elliptic curves in class 25410.ct have rank \(1\).

Complex multiplication

The elliptic curves in class 25410.ct do not have complex multiplication.

Modular form 25410.2.a.ct

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.