Properties

Label 177870dy
Number of curves $8$
Conductor $177870$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("dy1")
 
E.isogeny_class()
 

Elliptic curves in class 177870dy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
177870.ff7 177870dy1 \([1, 1, 1, -2949801, 1948034199]\) \(13619385906841/6048000\) \(1260538554778272000\) \([2]\) \(6635520\) \(2.4317\) \(\Gamma_0(N)\)-optimal
177870.ff6 177870dy2 \([1, 1, 1, -3424121, 1278863543]\) \(21302308926361/8930250000\) \(1861263959789792250000\) \([2, 2]\) \(13271040\) \(2.7783\)  
177870.ff5 177870dy3 \([1, 1, 1, -8730576, -7549690191]\) \(353108405631241/86318776320\) \(17990764806984412692480\) \([2]\) \(19906560\) \(2.9810\)  
177870.ff8 177870dy4 \([1, 1, 1, 11398379, 9407522543]\) \(785793873833639/637994920500\) \(-132972419815302337924500\) \([2]\) \(26542080\) \(3.1249\)  
177870.ff4 177870dy5 \([1, 1, 1, -25835741, -49667231041]\) \(9150443179640281/184570312500\) \(38468583825020507812500\) \([2]\) \(26542080\) \(3.1249\)  
177870.ff2 177870dy6 \([1, 1, 1, -130156496, -571548803407]\) \(1169975873419524361/108425318400\) \(22598262922835645337600\) \([2, 2]\) \(39813120\) \(3.3276\)  
177870.ff3 177870dy7 \([1, 1, 1, -120670096, -658383514447]\) \(-932348627918877961/358766164249920\) \(-74774897848369429827842880\) \([2]\) \(79626240\) \(3.6742\)  
177870.ff1 177870dy8 \([1, 1, 1, -2082457616, -36578228819791]\) \(4791901410190533590281/41160000\) \(8578665164463240000\) \([2]\) \(79626240\) \(3.6742\)  

Rank

sage: E.rank()
 

The elliptic curves in class 177870dy have rank \(0\).

Complex multiplication

The elliptic curves in class 177870dy do not have complex multiplication.

Modular form 177870.2.a.dy

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - q^{12} + 2 q^{13} + q^{15} + q^{16} - 6 q^{17} + q^{18} + 8 q^{19} + O(q^{20})\) Copy content 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.