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Properties

Label 112710cj
Number of curves $8$
Conductor $112710$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cj1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 112710cj have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(13\)\(1 - T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 112710cj do not have complex multiplication.

Modular form 112710.2.a.cj

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 4 q^{7} + q^{8} + q^{9} + q^{10} - q^{12} + q^{13} + 4 q^{14} - q^{15} + q^{16} + q^{18} - 4 q^{19} + O(q^{20})\) 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 112710cj

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
112710.cj7 112710cj1 \([1, 1, 1, -605786200, 5525820002585]\) \(1018563973439611524445729/42904970360310988800\) \(1035621682514961353618227200\) \([4]\) \(79626240\) \(3.9483\) \(\Gamma_0(N)\)-optimal
112710.cj6 112710cj2 \([1, 1, 1, -1606049880, -17425830293223]\) \(18980483520595353274840609/5549773448629762560000\) \(133958039550668849245616640000\) \([2, 2]\) \(159252480\) \(4.2949\)  
112710.cj5 112710cj3 \([1, 1, 1, -7454115160, -246099036985063]\) \(1897660325010178513043539489/14258428094958372000000\) \(344163791973596256277668000000\) \([4]\) \(238878720\) \(4.4977\)  
112710.cj8 112710cj4 \([1, 1, 1, 4300555240, -115815695740135]\) \(364421318680576777174674911/450962301637624725000000\) \(-10885133672176979795793525000000\) \([2]\) \(318504960\) \(4.6415\)  
112710.cj4 112710cj5 \([1, 1, 1, -23516873880, -1387921578504423]\) \(59589391972023341137821784609/8834417507562311995200\) \(213241362163593327583667668800\) \([2]\) \(318504960\) \(4.6415\)  
112710.cj2 112710cj6 \([1, 1, 1, -119049939240, -15810413259752295]\) \(7730680381889320597382223137569/441370202660156250000\) \(10653603721253505035156250000\) \([2, 2]\) \(477757440\) \(4.8442\)  
112710.cj3 112710cj7 \([1, 1, 1, -118834062020, -15870608118373543]\) \(-7688701694683937879808871873249/58423707246780395507812500\) \(-1410206264904961824417114257812500\) \([2]\) \(955514880\) \(5.1908\)  
112710.cj1 112710cj8 \([1, 1, 1, -1904799001740, -1011864096741752295]\) \(31664865542564944883878115208137569/103216295812500\) \(2491390462098629812500\) \([2]\) \(955514880\) \(5.1908\)