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Properties

Label 88200hd
Number of curves $6$
Conductor $88200$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("hd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 88200hd have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 88200hd do not have complex multiplication.

Modular form 88200.2.a.hd

Copy content sage:E.q_eigenform(10)
 
\(q - 4 q^{11} - 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 88200hd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
88200.s4 88200hd1 \([0, 0, 0, -8107050, 8884686125]\) \(2748251600896/2205\) \(47278574201250000\) \([2]\) \(2359296\) \(2.5050\) \(\Gamma_0(N)\)-optimal
88200.s3 88200hd2 \([0, 0, 0, -8162175, 8757733250]\) \(175293437776/4862025\) \(1667988097820100000000\) \([2, 2]\) \(4718592\) \(2.8516\)  
88200.s5 88200hd3 \([0, 0, 0, 1760325, 28692035750]\) \(439608956/259416045\) \(-355985726476983120000000\) \([2]\) \(9437184\) \(3.1982\)  
88200.s2 88200hd4 \([0, 0, 0, -18966675, -19301553250]\) \(549871953124/200930625\) \(275728644741690000000000\) \([2, 2]\) \(9437184\) \(3.1982\)  
88200.s6 88200hd5 \([0, 0, 0, 58208325, -136530378250]\) \(7947184069438/7533176175\) \(-20674921578859749600000000\) \([2]\) \(18874368\) \(3.5447\)  
88200.s1 88200hd6 \([0, 0, 0, -269013675, -1697867064250]\) \(784478485879202/221484375\) \(607867382587500000000000\) \([2]\) \(18874368\) \(3.5447\)