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Properties

Label 458640.d
Number of curves $6$
Conductor $458640$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 458640.d have rank \(0\).

Complex multiplication

The elliptic curves in class 458640.d do not have complex multiplication.

Modular form 458640.2.a.d

Copy content sage:E.q_eigenform(10)
 
\(q - q^{5} - 6 q^{11} - q^{13} + 6 q^{17} + 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 458640.d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.d1 458640d5 \([0, 0, 0, -6013641963, 179491755859738]\) \(68463752473882049153689/1817088000000000\) \(638339437673054208000000000\) \([2]\) \(429981696\) \(4.2487\) \(\Gamma_0(N)\)-optimal*
458640.d2 458640d6 \([0, 0, 0, -5778818283, 194153160179482]\) \(-60752633741424905775769/11197265625000000000\) \(-3933577373544000000000000000000\) \([2]\) \(859963392\) \(4.5952\)  
458640.d3 458640d3 \([0, 0, 0, -128832123, -161196398678]\) \(673163386034885929/357608625192000\) \(125627206118853510070272000\) \([2]\) \(143327232\) \(3.6994\) \(\Gamma_0(N)\)-optimal*
458640.d4 458640d1 \([0, 0, 0, -101207883, -391894428422]\) \(326355561310674169/465699780\) \(163599416038004244480\) \([2]\) \(47775744\) \(3.1501\) \(\Gamma_0(N)\)-optimal*
458640.d5 458640d2 \([0, 0, 0, -100290603, -399346594598]\) \(-317562142497484249/12339342574650\) \(-4334786757910050540134400\) \([2]\) \(95551488\) \(3.4966\)  
458640.d6 458640d4 \([0, 0, 0, 491249157, -1261096573142]\) \(37321015309599759191/23553520979625000\) \(-8274305557768422541824000000\) \([2]\) \(286654464\) \(4.0459\)  
*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 458640.d1.