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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("hq1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(7\)\(1\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 177870.2.a.hq

Copy content 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

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 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

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 177870hq

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
177870.bu8 177870hq1 \([1, 1, 0, 9421058, 1397530996]\) \(443688652450511/260789760000\) \(-54354422482039088640000\) \([2]\) \(19906560\) \(3.0519\) \(\Gamma_0(N)\)-optimal
177870.bu7 177870hq2 \([1, 1, 0, -38010942, 11178009396]\) \(29141055407581489/16604321025600\) \(3460712107917377499278400\) \([2, 2]\) \(39813120\) \(3.3984\)  
177870.bu6 177870hq3 \([1, 1, 0, -120068302, -556030258076]\) \(-918468938249433649/109183593750000\) \(-22756304476045464843750000\) \([2]\) \(59719680\) \(3.6012\)  
177870.bu4 177870hq4 \([1, 1, 0, -444740342, 3602679957276]\) \(46676570542430835889/106752955783320\) \(22249705125895331766315480\) \([2]\) \(79626240\) \(3.7450\)  
177870.bu5 177870hq5 \([1, 1, 0, -390193542, -2953424680884]\) \(31522423139920199089/164434491947880\) \(34271828180502655337761320\) \([2]\) \(79626240\) \(3.7450\)  
177870.bu3 177870hq6 \([1, 1, 0, -1972880802, -33729155820576]\) \(4074571110566294433649/48828650062500\) \(10176983462559148605562500\) \([2, 2]\) \(119439360\) \(3.9478\)  
177870.bu2 177870hq7 \([1, 1, 0, -2024759552, -31861738711326]\) \(4404531606962679693649/444872222400201750\) \(92721327428132989008782955750\) \([2]\) \(238878720\) \(4.2943\)  
177870.bu1 177870hq8 \([1, 1, 0, -31566002052, -2158639552679826]\) \(16689299266861680229173649/2396798250\) \(499546395858150044250\) \([2]\) \(238878720\) \(4.2943\)