Properties

Label 1290e
Number of curves $4$
Conductor $1290$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1290e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1290.c4 1290e1 \([1, 0, 1, 666, 8632]\) \(32740359775271/50724864000\) \(-50724864000\) \([2]\) \(1920\) \(0.73942\) \(\Gamma_0(N)\)-optimal
1290.c3 1290e2 \([1, 0, 1, -4454, 86456]\) \(9768641617435609/2396304000000\) \(2396304000000\) \([2, 2]\) \(3840\) \(1.0860\)  
1290.c2 1290e3 \([1, 0, 1, -24454, -1401544]\) \(1617141066657115609/89723013444000\) \(89723013444000\) \([2]\) \(7680\) \(1.4326\)  
1290.c1 1290e4 \([1, 0, 1, -66374, 6575672]\) \(32337636827233520089/3023437500000\) \(3023437500000\) \([2]\) \(7680\) \(1.4326\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1290e have rank \(1\).

Complex multiplication

The elliptic curves in class 1290e do not have complex multiplication.

Modular form 1290.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} + q^{10} + 4q^{11} + q^{12} - 6q^{13} - q^{15} + q^{16} - 6q^{17} - q^{18} - 8q^{19} + O(q^{20})\)  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.