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Properties

Label 72450dn
Number of curves $6$
Conductor $72450$
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, 28345, -3769153]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, 28345, -3769153]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, 28345, -3769153]) 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 72450dn have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1\)
\(7\)\(1 + T\)
\(23\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 13 T^{2}\) 1.13.a
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 6 T + 19 T^{2}\) 1.19.ag
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 72450dn do not have complex multiplication.

Modular form 72450.2.a.dn

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);
 
\(q + q^{2} + q^{4} - q^{7} + q^{8} + 4 q^{11} + 2 q^{13} - q^{14} + q^{16} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

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 72450dn

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
72450.dp5 72450dn1 \([1, -1, 1, 28345, -3769153]\) \(221115865823/664731648\) \(-7571708928000000\) \([2]\) \(524288\) \(1.7303\) \(\Gamma_0(N)\)-optimal
72450.dp4 72450dn2 \([1, -1, 1, -259655, -43513153]\) \(169967019783457/26337394944\) \(299999389284000000\) \([2, 2]\) \(1048576\) \(2.0769\)  
72450.dp3 72450dn3 \([1, -1, 1, -1141655, 427474847]\) \(14447092394873377/1439452851984\) \(16396267642130250000\) \([2, 2]\) \(2097152\) \(2.4234\)  
72450.dp2 72450dn4 \([1, -1, 1, -3985655, -3061573153]\) \(614716917569296417/19093020912\) \(217481441325750000\) \([2]\) \(2097152\) \(2.4234\)  
72450.dp6 72450dn5 \([1, -1, 1, 1409845, 2065537847]\) \(27207619911317663/177609314617308\) \(-2023081099312773937500\) \([2]\) \(4194304\) \(2.7700\)  
72450.dp1 72450dn6 \([1, -1, 1, -17805155, 28922059847]\) \(54804145548726848737/637608031452\) \(7262753983257937500\) \([2]\) \(4194304\) \(2.7700\)