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Properties

Label 18515a
Number of curves $2$
Conductor $18515$
CM no
Rank $2$
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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 18515a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
18515.k2 18515a1 \([0, 1, 1, -61, -184]\) \(48234496/6125\) \(3240125\) \([]\) \(3264\) \(-0.021614\) \(\Gamma_0(N)\)-optimal
18515.k1 18515a2 \([0, 1, 1, -1211, 15801]\) \(371585744896/588245\) \(311181605\) \([]\) \(9792\) \(0.52769\)  

Rank

sage: E.rank()
 

The elliptic curves in class 18515a have rank \(2\).

Complex multiplication

The elliptic curves in class 18515a do not have complex multiplication.

Modular form 18515.2.a.a

sage: E.q_eigenform(10)
 
\(q - 2 q^{3} - 2 q^{4} - q^{5} - q^{7} + q^{9} + 3 q^{11} + 4 q^{12} - 4 q^{13} + 2 q^{15} + 4 q^{16} - 6 q^{17} - 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.