Properties

Label 1078.e
Number of curves $2$
Conductor $1078$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1078.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1078.e1 1078a2 \([1, 0, 1, -271, -1718]\) \(911871625/10648\) \(25565848\) \([]\) \(288\) \(0.23367\)  
1078.e2 1078a1 \([1, 0, 1, -26, 46]\) \(765625/22\) \(52822\) \([3]\) \(96\) \(-0.31563\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1078.e have rank \(1\).

Complex multiplication

The elliptic curves in class 1078.e do not have complex multiplication.

Modular form 1078.2.a.e

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.