Properties

Label 10304.s
Number of curves $4$
Conductor $10304$
CM no
Rank $2$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("s1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 10304.s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10304.s1 10304h3 \([0, 0, 0, -7916, 271024]\) \(209267191953/55223\) \(14476378112\) \([2]\) \(10240\) \(0.93409\)  
10304.s2 10304h2 \([0, 0, 0, -556, 3120]\) \(72511713/25921\) \(6795034624\) \([2, 2]\) \(5120\) \(0.58751\)  
10304.s3 10304h1 \([0, 0, 0, -236, -1360]\) \(5545233/161\) \(42205184\) \([2]\) \(2560\) \(0.24094\) \(\Gamma_0(N)\)-optimal
10304.s4 10304h4 \([0, 0, 0, 1684, 21936]\) \(2014698447/1958887\) \(-513510473728\) \([2]\) \(10240\) \(0.93409\)  

Rank

sage: E.rank()
 

The elliptic curves in class 10304.s have rank \(2\).

Complex multiplication

The elliptic curves in class 10304.s do not have complex multiplication.

Modular form 10304.2.a.s

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.