Properties

Label 311696p
Number of curves $2$
Conductor $311696$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 311696p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
311696.p2 311696p1 \([0, 1, 0, -318512, 69082660]\) \(1969910093092/7889\) \(14311265002496\) \([2]\) \(1344000\) \(1.7357\) \(\Gamma_0(N)\)-optimal
311696.p1 311696p2 \([0, 1, 0, -323352, 66869812]\) \(1030541881826/62236321\) \(225803139209381888\) \([2]\) \(2688000\) \(2.0823\)  

Rank

sage: E.rank()
 

The elliptic curves in class 311696p have rank \(1\).

Complex multiplication

The elliptic curves in class 311696p do not have complex multiplication.

Modular form 311696.2.a.p

sage: E.q_eigenform(10)
 
\(q - 2 q^{3} + 2 q^{5} + q^{7} + q^{9} - 4 q^{13} - 4 q^{15} + 2 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.