↑

Properties

Label 456300q
Number of curves $2$
Conductor $456300$
CM no
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, 12675, 274625]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, 12675, 274625]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, 12675, 274625]) 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 456300q have rank \(1\).

Complex multiplication

The elliptic curves in class 456300q do not have complex multiplication.

Modular form 456300.2.a.q

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);
 
\(q - 4 q^{7} + 6 q^{11} - 3 q^{17} + 7 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 456300q

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
456300.q2 456300q1 \([0, 0, 0, 12675, 274625]\) \(6912/5\) \(-162904803750000\) \([]\) \(1866240\) \(1.4154\) \(\Gamma_0(N)\)-optimal*
456300.q1 456300q2 \([0, 0, 0, -240825, 46411625]\) \(-5267712/125\) \(-36653580843750000\) \([]\) \(5598720\) \(1.9647\) \(\Gamma_0(N)\)-optimal*
*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 456300q1.