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Properties

Label 327600.z
Number of curves $6$
Conductor $327600$
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([0, 0, 0, -220194075, -1257640429750]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -220194075, -1257640429750]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -220194075, -1257640429750]) 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 327600.z have rank \(1\).

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
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 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 327600.z do not have complex multiplication.

Modular form 327600.2.a.z

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);
 
\(q - q^{7} - 4 q^{11} - q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB 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 comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

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

Elliptic curves in class 327600.z

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
327600.z1 327600z6 \([0, 0, 0, -220194075, -1257640429750]\) \(25306558948218234961/4478906250\) \(208967850000000000000\) \([2]\) \(37748736\) \(3.2956\)  
327600.z2 327600z4 \([0, 0, 0, -13806075, -19518817750]\) \(6237734630203441/82168222500\) \(3833640588960000000000\) \([2, 2]\) \(18874368\) \(2.9490\)  
327600.z3 327600z5 \([0, 0, 0, -2106075, -51541717750]\) \(-22143063655441/24584858584650\) \(-1147031162125430400000000\) \([2]\) \(37748736\) \(3.2956\)  
327600.z4 327600z2 \([0, 0, 0, -1638075, 327190250]\) \(10418796526321/5038160400\) \(235060411622400000000\) \([2, 2]\) \(9437184\) \(2.6025\)  
327600.z5 327600z1 \([0, 0, 0, -1350075, 603382250]\) \(5832972054001/4542720\) \(211945144320000000\) \([2]\) \(4718592\) \(2.2559\) \(\Gamma_0(N)\)-optimal
327600.z6 327600z3 \([0, 0, 0, 5921925, 2496910250]\) \(492271755328079/342606902820\) \(-15984667657969920000000\) \([2]\) \(18874368\) \(2.9490\)