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Properties

Label 425880.bj
Number of curves $2$
Conductor $425880$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -27838863, 56305476162]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -27838863, 56305476162]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -27838863, 56305476162]) 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 425880.bj have rank \(0\).

Complex multiplication

The elliptic curves in class 425880.bj do not have complex multiplication.

Modular form 425880.2.a.bj

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);
 
\(q - q^{5} + q^{7} - 6 q^{11} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

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 425880.bj

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
425880.bj1 425880bj2 \([0, 0, 0, -27838863, 56305476162]\) \(98104024066032/462109375\) \(11239219447010100000000\) \([2]\) \(45416448\) \(3.0808\) \(\Gamma_0(N)\)-optimal*
425880.bj2 425880bj1 \([0, 0, 0, -848718, 1779985233]\) \(-44477724672/874680625\) \(-1329599660581294830000\) \([2]\) \(22708224\) \(2.7342\) \(\Gamma_0(N)\)-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 425880.bj1.