↑

Properties

Label 41745.h
Number of curves $4$
Conductor $41745$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("h1")
 
E.isogeny_class()
 

Elliptic curves in class 41745.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
41745.h1 41745z4 \([1, 0, 0, -3646156, -2679050989]\) \(3026030815665395929/1364501953125\) \(2417298444580078125\) \([2]\) \(1536000\) \(2.4852\)  
41745.h2 41745z3 \([1, 0, 0, -2004186, 1072928385]\) \(502552788401502649/10024505152875\) \(17759022373132387875\) \([2]\) \(1536000\) \(2.4852\)  
41745.h3 41745z2 \([1, 0, 0, -264811, -27400240]\) \(1159246431432649/488076890625\) \(864657984432515625\) \([2, 2]\) \(768000\) \(2.1386\)  
41745.h4 41745z1 \([1, 0, 0, 55234, -3140829]\) \(10519294081031/8500170375\) \(-15058570329705375\) \([2]\) \(384000\) \(1.7921\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 41745.h have rank \(1\).

Complex multiplication

The elliptic curves in class 41745.h do not have complex multiplication.

Modular form 41745.2.a.h

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.