↑

Properties

Label 85782.m
Number of curves $4$
Conductor $85782$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 85782.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
85782.m1 85782m3 \([1, 1, 1, -631188, -146187435]\) \(46753267515625/11591221248\) \(6894728717181124608\) \([2]\) \(1814400\) \(2.3251\)  
85782.m2 85782m1 \([1, 1, 1, -214893, 38238819]\) \(1845026709625/793152\) \(471785306697792\) \([2]\) \(604800\) \(1.7758\) \(\Gamma_0(N)\)-optimal
85782.m3 85782m2 \([1, 1, 1, -181253, 50658707]\) \(-1107111813625/1228691592\) \(-730854413238217032\) \([2]\) \(1209600\) \(2.1223\)  
85782.m4 85782m4 \([1, 1, 1, 1521772, -924697771]\) \(655215969476375/1001033261568\) \(-595437929077339427328\) \([2]\) \(3628800\) \(2.6716\)  

Rank

sage: E.rank()
 

The elliptic curves in class 85782.m have rank \(1\).

Complex multiplication

The elliptic curves in class 85782.m do not have complex multiplication.

Modular form 85782.2.a.m

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

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.