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Properties

Label 33600.hf
Number of curves $8$
Conductor $33600$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("hf1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 33600.hf have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 8 T + 23 T^{2}\) 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 33600.hf do not have complex multiplication.

Modular form 33600.2.a.hf

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + q^{7} + q^{9} + 4 q^{11} - 2 q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 33600.hf

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
33600.hf1 33600da8 \([0, 1, 0, -3073280033, -65577989759937]\) \(783736670177727068275201/360150\) \(1475174400000000\) \([2]\) \(9437184\) \(3.6389\)  
33600.hf2 33600da6 \([0, 1, 0, -192080033, -1024703759937]\) \(191342053882402567201/129708022500\) \(531284060160000000000\) \([2, 2]\) \(4718592\) \(3.2923\)  
33600.hf3 33600da7 \([0, 1, 0, -190880033, -1038137759937]\) \(-187778242790732059201/4984939585440150\) \(-20418312541962854400000000\) \([2]\) \(9437184\) \(3.6389\)  
33600.hf4 33600da4 \([0, 1, 0, -24112033, 45551504063]\) \(378499465220294881/120530818800\) \(493694233804800000000\) \([2]\) \(2359296\) \(2.9457\)  
33600.hf5 33600da3 \([0, 1, 0, -12080033, -15803759937]\) \(47595748626367201/1215506250000\) \(4978713600000000000000\) \([2, 2]\) \(2359296\) \(2.9457\)  
33600.hf6 33600da2 \([0, 1, 0, -1712033, 505104063]\) \(135487869158881/51438240000\) \(210691031040000000000\) \([2, 2]\) \(1179648\) \(2.5991\)  
33600.hf7 33600da1 \([0, 1, 0, 335967, 56592063]\) \(1023887723039/928972800\) \(-3805072588800000000\) \([2]\) \(589824\) \(2.2526\) \(\Gamma_0(N)\)-optimal
33600.hf8 33600da5 \([0, 1, 0, 2031967, -50505167937]\) \(226523624554079/269165039062500\) \(-1102500000000000000000000\) \([2]\) \(4718592\) \(3.2923\)