↑

Properties

Label 494209i
Number of curves $2$
Conductor $494209$
CM \(\Q(\sqrt{-19}) \)
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve([0, 0, 1, -52022, 4571433]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 494209i have rank \(1\).

Complex multiplication

Each elliptic curve in class 494209i has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-19}) \).

Modular form 494209.2.a.i

Copy content sage:E.q_eigenform(10)
 
\(q - 2 q^{4} + q^{5} + 3 q^{7} - 3 q^{9} - 5 q^{11} + 4 q^{16} + 7 q^{17} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

\(\left(\begin{array}{rr} 1 & 19 \\ 19 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 494209i

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
494209.i2 494209i1 \([0, 0, 1, -52022, 4571433]\) \(-884736\) \(-17598317439331\) \([]\) \(1039680\) \(1.4553\) \(\Gamma_0(N)\)-optimal \(-19\)
494209.i1 494209i2 \([0, 0, 1, -18779942, -31355460662]\) \(-884736\) \(-827928348050990945611\) \([]\) \(19753920\) \(2.9275\)   \(-19\)