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Properties

Label 428400.bj
Number of curves $4$
Conductor $428400$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bj1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 428400.bj have rank \(0\).

Complex multiplication

The elliptic curves in class 428400.bj do not have complex multiplication.

Modular form 428400.2.a.bj

Copy content sage:E.q_eigenform(10)
 
\(q - q^{7} - 4 q^{11} - 2 q^{13} + q^{17} + 8 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 428400.bj

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
428400.bj1 428400bj3 \([0, 0, 0, -4971771675, 134931974978250]\) \(291306206119284545407569/101150000000\) \(4719254400000000000000\) \([2]\) \(198180864\) \(3.9524\) \(\Gamma_0(N)\)-optimal*
428400.bj2 428400bj4 \([0, 0, 0, -368379675, 1271852354250]\) \(118495863754334673489/53596139570691200\) \(2500581487810168627200000000\) \([2]\) \(198180864\) \(3.9524\)  
428400.bj3 428400bj2 \([0, 0, 0, -310779675, 2107685954250]\) \(71149857462630609489/41907496960000\) \(1955236178165760000000000\) \([2, 2]\) \(99090432\) \(3.6058\) \(\Gamma_0(N)\)-optimal*
428400.bj4 428400bj1 \([0, 0, 0, -15867675, 45366338250]\) \(-9470133471933009/13576123187200\) \(-633407603422003200000000\) \([2]\) \(49545216\) \(3.2592\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 428400.bj1.