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Properties

Label 327600hm
Number of curves $6$
Conductor $327600$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("hm1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 327600hm have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1\)
\(7\)\(1 - T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 327600hm do not have complex multiplication.

Modular form 327600.2.a.hm

Copy content sage:E.q_eigenform(10)
 
\(q + q^{7} - 4 q^{11} - q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 327600hm

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
327600.hm5 327600hm1 \([0, 0, 0, -1332075, 9322852250]\) \(-5602762882081/801531494400\) \(-37396253402726400000000\) \([2]\) \(28311552\) \(3.0107\) \(\Gamma_0(N)\)-optimal
327600.hm4 327600hm2 \([0, 0, 0, -75060075, 248275300250]\) \(1002404925316922401/9348917760000\) \(436183107010560000000000\) \([2, 2]\) \(56623104\) \(3.3573\)  
327600.hm2 327600hm3 \([0, 0, 0, -1198260075, 15965212900250]\) \(4078208988807294650401/359723582400\) \(16783263460454400000000\) \([2]\) \(113246208\) \(3.7039\)  
327600.hm3 327600hm4 \([0, 0, 0, -131508075, -175705627750]\) \(5391051390768345121/2833965225000000\) \(132221481537600000000000000\) \([2, 2]\) \(113246208\) \(3.7039\)  
327600.hm6 327600hm5 \([0, 0, 0, 498491925, -1370815627750]\) \(293623352309352854879/187320324116835000\) \(-8739617041995053760000000000\) \([2]\) \(226492416\) \(4.0504\)  
327600.hm1 327600hm6 \([0, 0, 0, -1664676075, -26115375019750]\) \(10934663514379917006241/12996826171875000\) \(606379921875000000000000000\) \([2]\) \(226492416\) \(4.0504\)