Properties

Label 51870b
Number of curves $4$
Conductor $51870$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 51870b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51870.e4 51870b1 \([1, 1, 0, -14213, -690483]\) \(-317562142497484249/18670942617600\) \(-18670942617600\) \([2]\) \(163840\) \(1.3024\) \(\Gamma_0(N)\)-optimal
51870.e3 51870b2 \([1, 1, 0, -230533, -42699827]\) \(1354958399265695661529/4304795040000\) \(4304795040000\) \([2, 2]\) \(327680\) \(1.6490\)  
51870.e2 51870b3 \([1, 1, 0, -233653, -41488643]\) \(1410719602237262088409/76269550743750000\) \(76269550743750000\) \([2]\) \(655360\) \(1.9955\)  
51870.e1 51870b4 \([1, 1, 0, -3688533, -2728182627]\) \(5549896908024170183373529/56019600\) \(56019600\) \([2]\) \(655360\) \(1.9955\)  

Rank

sage: E.rank()
 

The elliptic curves in class 51870b have rank \(0\).

Complex multiplication

The elliptic curves in class 51870b do not have complex multiplication.

Modular form 51870.2.a.b

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.