↑

Properties

Label 435600.z
Number of curves $2$
Conductor $435600$
CM \(\Q(\sqrt{-3}) \)
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, 0, -73205]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, 0, -73205]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, 0, -73205]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 435600.z have rank \(1\).

Complex multiplication

Each elliptic curve in class 435600.z has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 435600.2.a.z

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - 4 q^{7} - 5 q^{13} + 7 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 435600.z

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
435600.z1 435600z1 \([0, 0, 0, 0, -73205]\) \(0\) \(-2315075914800\) \([]\) \(684288\) \(1.0514\) \(\Gamma_0(N)\)-optimal* \(-3\)
435600.z2 435600z2 \([0, 0, 0, 0, 1976535]\) \(0\) \(-1687690341889200\) \([]\) \(2052864\) \(1.6007\) \(\Gamma_0(N)\)-optimal* \(-3\)
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 435600.z1.