Properties

Label 10304.e
Number of curves $2$
Conductor $10304$
CM no
Rank $1$
Graph

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

Elliptic curves in class 10304.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10304.e1 10304p2 \([0, 1, 0, -10689, 399007]\) \(1030541881826/62236321\) \(8157439066112\) \([2]\) \(15360\) \(1.2299\)  
10304.e2 10304p1 \([0, 1, 0, -10529, 412351]\) \(1969910093092/7889\) \(517013504\) \([2]\) \(7680\) \(0.88336\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 10304.2.a.e

sage: E.q_eigenform(10)
 
\(q - 2q^{3} - 2q^{5} + q^{7} + q^{9} - 2q^{11} - 4q^{13} + 4q^{15} - 2q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.