Properties

Label 31824r
Number of curves $2$
Conductor $31824$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 31824r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31824.t2 31824r1 \([0, 0, 0, -4755, -119662]\) \(3981876625/232713\) \(694877294592\) \([2]\) \(32768\) \(1.0258\) \(\Gamma_0(N)\)-optimal
31824.t1 31824r2 \([0, 0, 0, -14115, 496226]\) \(104154702625/24649677\) \(73603541127168\) \([2]\) \(65536\) \(1.3724\)  

Rank

sage: E.rank()
 

The elliptic curves in class 31824r have rank \(1\).

Complex multiplication

The elliptic curves in class 31824r do not have complex multiplication.

Modular form 31824.2.a.r

sage: E.q_eigenform(10)
 
\(q + 2 q^{7} - 2 q^{11} - q^{13} - q^{17} + O(q^{20})\) 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.