Properties

Label 1050k
Number of curves $8$
Conductor $1050$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1050k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1050.k7 1050k1 \([1, 1, 1, -12438, 528531]\) \(13619385906841/6048000\) \(94500000000\) \([4]\) \(2304\) \(1.0645\) \(\Gamma_0(N)\)-optimal
1050.k6 1050k2 \([1, 1, 1, -14438, 344531]\) \(21302308926361/8930250000\) \(139535156250000\) \([2, 2]\) \(4608\) \(1.4111\)  
1050.k5 1050k3 \([1, 1, 1, -36813, -2081469]\) \(353108405631241/86318776320\) \(1348730880000000\) \([4]\) \(6912\) \(1.6138\)  
1050.k4 1050k4 \([1, 1, 1, -108938, -13641469]\) \(9150443179640281/184570312500\) \(2883911132812500\) \([2]\) \(9216\) \(1.7577\)  
1050.k8 1050k5 \([1, 1, 1, 48062, 2594531]\) \(785793873833639/637994920500\) \(-9968670632812500\) \([2]\) \(9216\) \(1.7577\)  
1050.k2 1050k6 \([1, 1, 1, -548813, -156705469]\) \(1169975873419524361/108425318400\) \(1694145600000000\) \([2, 2]\) \(13824\) \(1.9604\)  
1050.k1 1050k7 \([1, 1, 1, -8780813, -10018641469]\) \(4791901410190533590281/41160000\) \(643125000000\) \([2]\) \(27648\) \(2.3070\)  
1050.k3 1050k8 \([1, 1, 1, -508813, -180465469]\) \(-932348627918877961/358766164249920\) \(-5605721316405000000\) \([2]\) \(27648\) \(2.3070\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1050k have rank \(0\).

Complex multiplication

The elliptic curves in class 1050k do not have complex multiplication.

Modular form 1050.2.a.k

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} + q^{9} - q^{12} - 2q^{13} - q^{14} + q^{16} + 6q^{17} + q^{18} + 8q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.