Properties

Label 4410.bi
Number of curves $8$
Conductor $4410$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("bi1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4410.bi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4410.bi1 4410bk7 \([1, -1, 1, -154893542, -741951322059]\) \(4791901410190533590281/41160000\) \(3530133540360000\) \([2]\) \(442368\) \(3.0245\)  
4410.bi2 4410bk6 \([1, -1, 1, -9681062, -11590632651]\) \(1169975873419524361/108425318400\) \(9299218977357926400\) \([2, 2]\) \(221184\) \(2.6780\)  
4410.bi3 4410bk8 \([1, -1, 1, -8975462, -13352374731]\) \(-932348627918877961/358766164249920\) \(-30769982253764512960320\) \([2]\) \(442368\) \(3.0245\)  
4410.bi4 4410bk4 \([1, -1, 1, -1921667, -1006825809]\) \(9150443179640281/184570312500\) \(15829879754882812500\) \([2]\) \(147456\) \(2.4752\)  
4410.bi5 4410bk3 \([1, -1, 1, -649382, -152913099]\) \(353108405631241/86318776320\) \(7403226614433054720\) \([4]\) \(110592\) \(2.3314\)  
4410.bi6 4410bk2 \([1, -1, 1, -254687, 26034999]\) \(21302308926361/8930250000\) \(765912902060250000\) \([2, 2]\) \(73728\) \(2.1286\)  
4410.bi7 4410bk1 \([1, -1, 1, -219407, 39596631]\) \(13619385906841/6048000\) \(518713499808000\) \([4]\) \(36864\) \(1.7821\) \(\Gamma_0(N)\)-optimal
4410.bi8 4410bk5 \([1, -1, 1, 847813, 190527999]\) \(785793873833639/637994920500\) \(-54718349548988380500\) \([2]\) \(147456\) \(2.4752\)  

Rank

sage: E.rank()
 

The elliptic curves in class 4410.bi have rank \(1\).

Complex multiplication

The elliptic curves in class 4410.bi do not have complex multiplication.

Modular form 4410.2.a.bi

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} - 2q^{13} + q^{16} - 6q^{17} - 8q^{19} + O(q^{20})\)  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.