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Properties

Label 456300c
Number of curves $2$
Conductor $456300$
CM \(\Q(\sqrt{-3}) \)
Rank $2$
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, 0, 54925]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, 0, 54925]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, 0, 54925]) 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 456300c have rank \(2\).

Complex multiplication

Each elliptic curve in class 456300c has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 456300.2.a.c

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);
 
\(q - 5 q^{7} + q^{19} + O(q^{20})\) 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 456300c

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
456300.c2 456300c1 \([0, 0, 0, 0, 54925]\) \(0\) \(-1303238430000\) \([]\) \(746496\) \(1.0035\) \(\Gamma_0(N)\)-optimal \(-3\)
456300.c1 456300c2 \([0, 0, 0, 0, -1482975]\) \(0\) \(-950060815470000\) \([]\) \(2239488\) \(1.5528\)   \(-3\)