Properties

Label 214774e
Number of curves $4$
Conductor $214774$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 214774e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
214774.o4 214774e1 \([1, -1, 1, -23814886, -45137954955]\) \(-10090256344188054273/107965577101312\) \(-15982780187590764986368\) \([2]\) \(20275200\) \(3.0768\) \(\Gamma_0(N)\)-optimal
214774.o3 214774e2 \([1, -1, 1, -382011366, -2873743918219]\) \(41647175116728660507393/4693358285056\) \(694785466123780374784\) \([2, 2]\) \(40550400\) \(3.4234\)  
214774.o2 214774e3 \([1, -1, 1, -382984726, -2858362494155]\) \(41966336340198080824833/442001722607124848\) \(65432117945677124607669872\) \([4]\) \(81100800\) \(3.7700\)  
214774.o1 214774e4 \([1, -1, 1, -6112181686, -183924205348939]\) \(170586815436843383543017473/2166416\) \(320707318503824\) \([2]\) \(81100800\) \(3.7700\)  

Rank

sage: E.rank()
 

The elliptic curves in class 214774e have rank \(1\).

Complex multiplication

The elliptic curves in class 214774e do not have complex multiplication.

Modular form 214774.2.a.e

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - 2 q^{5} + q^{7} + q^{8} - 3 q^{9} - 2 q^{10} - 4 q^{11} + 6 q^{13} + q^{14} + q^{16} - 6 q^{17} - 3 q^{18} - 4 q^{19} + O(q^{20})\)  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 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.