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Properties

Label 258570ef
Number of curves $8$
Conductor $258570$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, -3188238098, -66720369131503]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -3188238098, -66720369131503]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -3188238098, -66720369131503]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 258570ef have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(13\)\(1\)
\(17\)\(1 + T\)
 
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.ae
\(23\) \( 1 + 23 T^{2}\) 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 258570ef do not have complex multiplication.

Modular form 258570.2.a.ef

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q + q^{2} + q^{4} - q^{5} + 4 q^{7} + q^{8} - q^{10} + 4 q^{14} + q^{16} - q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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 comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 258570ef

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
258570.ef7 258570ef1 \([1, -1, 1, -3188238098, -66720369131503]\) \(1018563973439611524445729/42904970360310988800\) \(150971596771234213859740876800\) \([2]\) \(371589120\) \(4.3635\) \(\Gamma_0(N)\)-optimal
258570.ef6 258570ef2 \([1, -1, 1, -8452601618, 210391515070481]\) \(18980483520595353274840609/5549773448629762560000\) \(19528230697329431006911388160000\) \([2, 2]\) \(743178240\) \(4.7101\)  
258570.ef5 258570ef3 \([1, -1, 1, -39230827538, 2971343928843281]\) \(1897660325010178513043539489/14258428094958372000000\) \(50171754900801887029717092000000\) \([2]\) \(1114767360\) \(4.9128\)  
258570.ef4 258570ef4 \([1, -1, 1, -123768737618, 16757519007800081]\) \(59589391972023341137821784609/8834417507562311995200\) \(31086051486804055651845461947200\) \([2]\) \(1486356480\) \(5.0567\)  
258570.ef8 258570ef5 \([1, -1, 1, 22633718062, 1398361438905617]\) \(364421318680576777174674911/450962301637624725000000\) \(-1586820785333592083953090725000000\) \([2]\) \(1486356480\) \(5.0567\)  
258570.ef2 258570ef6 \([1, -1, 1, -626556946658, 190892678863373777]\) \(7730680381889320597382223137569/441370202660156250000\) \(1553068646901730407972656250000\) \([2, 2]\) \(2229534720\) \(5.2594\)  
258570.ef1 258570ef7 \([1, -1, 1, -10024911009158, 12217124280120623777]\) \(31664865542564944883878115208137569/103216295812500\) \(363191696923764800812500\) \([2]\) \(4459069440\) \(5.6060\)  
258570.ef3 258570ef8 \([1, -1, 1, -625420790078, 191619465502646081]\) \(-7688701694683937879808871873249/58423707246780395507812500\) \(-205578055369099004030227661132812500\) \([2]\) \(4459069440\) \(5.6060\)