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Properties

Label 487872.mk
Number of curves $4$
Conductor $487872$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("mk1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 487872.mk have rank \(1\).

Complex multiplication

The elliptic curves in class 487872.mk do not have complex multiplication.

Modular form 487872.2.a.mk

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{5} - q^{7} - 2 q^{13} + 6 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 487872.mk

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
487872.mk1 487872mk4 \([0, 0, 0, -26102604, -51056244272]\) \(46477380430664/286446699\) \(12122086984393828958208\) \([2]\) \(35389440\) \(3.0757\)  
487872.mk2 487872mk2 \([0, 0, 0, -2623764, 287283040]\) \(377619516352/211789809\) \(1120336914366987964416\) \([2, 2]\) \(17694720\) \(2.7292\)  
487872.mk3 487872mk1 \([0, 0, 0, -1964919, 1058395228]\) \(10150654719808/19370043\) \(1601010565929770688\) \([2]\) \(8847360\) \(2.3826\) \(\Gamma_0(N)\)-optimal*
487872.mk4 487872mk3 \([0, 0, 0, 10313556, 2279630320]\) \(2866919053816/1712145897\) \(-72455998152057591988224\) \([2]\) \(35389440\) \(3.0757\)  
*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 487872.mk1.