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Properties

Label 327600.fp
Number of curves $4$
Conductor $327600$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fp1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 327600.fp 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.ae
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 327600.2.a.fp

Copy content sage:E.q_eigenform(10)
 
\(q - q^{7} + 4 q^{11} - q^{13} - 6 q^{17} + 8 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 327600.fp

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
327600.fp1 327600fp3 \([0, 0, 0, -567844428675, 164699456938789250]\) \(434014578033107719741685694649/103121648659575000\) \(4811243639861131200000000000\) \([2]\) \(2123366400\) \(5.1303\)  
327600.fp2 327600fp2 \([0, 0, 0, -35494428675, 2572796788789250]\) \(105997782562506306791694649/51649016225625000000\) \(2409736501022760000000000000000\) \([2, 2]\) \(1061683200\) \(4.7837\)  
327600.fp3 327600fp4 \([0, 0, 0, -29580780675, 3458205908293250]\) \(-61354313914516350666047929/75227254486083984375000\) \(-3509802785302734375000000000000000\) \([2]\) \(2123366400\) \(5.1303\)  
327600.fp4 327600fp1 \([0, 0, 0, -2592156675, 25733206453250]\) \(41285728533151645510969/17760741842188800000\) \(828645171389160652800000000000\) \([2]\) \(530841600\) \(4.4372\) \(\Gamma_0(N)\)-optimal