Properties

Label 333270b
Number of curves $4$
Conductor $333270$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 333270b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.b4 333270b1 \([1, -1, 0, -8765100, 9947610000]\) \(690080604747409/3406760000\) \(367651281257827560000\) \([2]\) \(31629312\) \(2.7938\) \(\Gamma_0(N)\)-optimal
333270.b3 333270b2 \([1, -1, 0, -13526100, -2047253400]\) \(2535986675931409/1450751712200\) \(156562459867239577288200\) \([2]\) \(63258624\) \(3.1404\)  
333270.b2 333270b3 \([1, -1, 0, -50376240, -130429541844]\) \(131010595463836369/7704101562500\) \(831412488814461914062500\) \([2]\) \(94887936\) \(3.3431\)  
333270.b1 333270b4 \([1, -1, 0, -794282490, -8615870573094]\) \(513516182162686336369/1944885031250\) \(209888419976233281281250\) \([2]\) \(189775872\) \(3.6897\)  

Rank

sage: E.rank()
 

The elliptic curves in class 333270b have rank \(1\).

Complex multiplication

The elliptic curves in class 333270b do not have complex multiplication.

Modular form 333270.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + q^{10} - 6 q^{11} - 4 q^{13} + q^{14} + q^{16} + 6 q^{17} - 8 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.