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Properties

Label 327795d
Number of curves $8$
Conductor $327795$
CM no
Rank $2$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 327795d have rank \(2\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(13\)\(1 + T\)
\(41\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 327795d do not have complex multiplication.

Modular form 327795.2.a.d

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3 q^{8} + q^{9} - q^{10} - 4 q^{11} + q^{12} - q^{13} - q^{15} - q^{16} - 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 327795d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
327795.d6 327795d1 \([1, 1, 1, -184945, 30535430]\) \(147281603041/5265\) \(25009298828865\) \([2]\) \(1658880\) \(1.6599\) \(\Gamma_0(N)\)-optimal
327795.d5 327795d2 \([1, 1, 1, -193350, 27597042]\) \(168288035761/27720225\) \(131673958333974225\) \([2, 2]\) \(3317760\) \(2.0065\)  
327795.d7 327795d3 \([1, 1, 1, 352975, 155874152]\) \(1023887723039/2798036865\) \(-13290966778910844465\) \([2]\) \(6635520\) \(2.3531\)  
327795.d4 327795d4 \([1, 1, 1, -874155, -288568800]\) \(15551989015681/1445900625\) \(6868178690877050625\) \([2, 2]\) \(6635520\) \(2.3531\)  
327795.d8 327795d5 \([1, 1, 1, 1016970, -1364240700]\) \(24487529386319/183539412225\) \(-871831340400619746225\) \([2]\) \(13271040\) \(2.6996\)  
327795.d2 327795d6 \([1, 1, 1, -13658160, -19433894688]\) \(59319456301170001/594140625\) \(2822229902562890625\) \([2, 2]\) \(13271040\) \(2.6996\)  
327795.d3 327795d7 \([1, 1, 1, -13330365, -20410592670]\) \(-55150149867714721/5950927734375\) \(-28267527068939208984375\) \([2]\) \(26542080\) \(3.0462\)  
327795.d1 327795d8 \([1, 1, 1, -218530035, -1243502373438]\) \(242970740812818720001/24375\) \(115783790874375\) \([2]\) \(26542080\) \(3.0462\)