Properties

Label 10304.r
Number of curves $4$
Conductor $10304$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 10304.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10304.r1 10304w3 \([0, 0, 0, -7916, -271024]\) \(209267191953/55223\) \(14476378112\) \([2]\) \(10240\) \(0.93409\)  
10304.r2 10304w2 \([0, 0, 0, -556, -3120]\) \(72511713/25921\) \(6795034624\) \([2, 2]\) \(5120\) \(0.58751\)  
10304.r3 10304w1 \([0, 0, 0, -236, 1360]\) \(5545233/161\) \(42205184\) \([2]\) \(2560\) \(0.24094\) \(\Gamma_0(N)\)-optimal
10304.r4 10304w4 \([0, 0, 0, 1684, -21936]\) \(2014698447/1958887\) \(-513510473728\) \([2]\) \(10240\) \(0.93409\)  

Rank

sage: E.rank()
 

The elliptic curves in class 10304.r have rank \(1\).

Complex multiplication

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

Modular form 10304.2.a.r

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.