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Properties

Label 478800.pj
Number of curves $6$
Conductor $478800$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("pj1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 478800.pj have rank \(0\).

Complex multiplication

The elliptic curves in class 478800.pj do not have complex multiplication.

Modular form 478800.2.a.pj

Copy content sage:E.q_eigenform(10)
 
\(q + q^{7} + 6 q^{11} + 4 q^{13} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 478800.pj

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
478800.pj1 478800pj6 \([0, 0, 0, -35232157875, -2545405502660750]\) \(103665426767620308239307625/5961940992\) \(278160318922752000000\) \([2]\) \(573308928\) \(4.3094\)  
478800.pj2 478800pj5 \([0, 0, 0, -2202013875, -39771808964750]\) \(25309080274342544331625/191933498523648\) \(8954849307119321088000000\) \([2]\) \(286654464\) \(3.9628\)  
478800.pj3 478800pj4 \([0, 0, 0, -435367875, -3484846322750]\) \(195607431345044517625/752875610010048\) \(35126164460628799488000000\) \([2]\) \(191102976\) \(3.7601\)  
478800.pj4 478800pj3 \([0, 0, 0, -40231875, 3019149250]\) \(154357248921765625/89242711068672\) \(4163707927619960832000000\) \([2]\) \(95551488\) \(3.4135\)  
478800.pj5 478800pj2 \([0, 0, 0, -28585875, 55114983250]\) \(55369510069623625/3916046302812\) \(182707056303996672000000\) \([2]\) \(63700992\) \(3.2108\)  
478800.pj6 478800pj1 \([0, 0, 0, -28081875, 57277647250]\) \(52492168638015625/293197968\) \(13679444395008000000\) \([2]\) \(31850496\) \(2.8642\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 478800.pj1.