Properties

Label 177870cz
Number of curves $8$
Conductor $177870$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cz1")
 
E.isogeny_class()
 

Elliptic curves in class 177870cz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
177870.gk6 177870cz1 \([1, 1, 1, -1698857245, 26950822819955]\) \(2601656892010848045529/56330588160\) \(11740555256120443146240\) \([4]\) \(79626240\) \(3.7617\) \(\Gamma_0(N)\)-optimal
177870.gk5 177870cz2 \([1, 1, 1, -1700754525, 26887606209267]\) \(2610383204210122997209/12104550027662400\) \(2522859126671768196973953600\) \([2, 2]\) \(159252480\) \(4.1083\)  
177870.gk4 177870cz3 \([1, 1, 1, -1812782980, 23129897849477]\) \(3160944030998056790089/720291785342976000\) \(150124928259738143098404864000\) \([4]\) \(238878720\) \(4.3110\)  
177870.gk7 177870cz4 \([1, 1, 1, -836306325, 54199674198627]\) \(-310366976336070130009/5909282337130963560\) \(-1231626689322723924873128556840\) \([2]\) \(318504960\) \(4.4548\)  
177870.gk3 177870cz5 \([1, 1, 1, -2595559205, -4470286918525]\) \(9278380528613437145689/5328033205714065000\) \(1110481361928149982083251785000\) \([2]\) \(318504960\) \(4.4548\)  
177870.gk2 177870cz6 \([1, 1, 1, -9584041860, -341360794643835]\) \(467116778179943012100169/28800309694464000000\) \(6002629093820487267127296000000\) \([2, 2]\) \(477757440\) \(4.6576\)  
177870.gk8 177870cz7 \([1, 1, 1, 7491478140, -1425396766739835]\) \(223090928422700449019831/4340371122724101696000\) \(-904630479867722388764801530944000\) \([2]\) \(955514880\) \(5.0041\)  
177870.gk1 177870cz8 \([1, 1, 1, -150999703940, -22584573716942203]\) \(1826870018430810435423307849/7641104625000000000\) \(1592577212451565811625000000000\) \([2]\) \(955514880\) \(5.0041\)  

Rank

sage: E.rank()
 

The elliptic curves in class 177870cz have rank \(1\).

Complex multiplication

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

Modular form 177870.2.a.cz

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