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Properties

Label 29400o
Number of curves $6$
Conductor $29400$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("o1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 29400o have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 29400o do not have complex multiplication.

Modular form 29400.2.a.o

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{9} + 4 q^{11} - 2 q^{13} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 29400o

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
29400.cc4 29400o1 \([0, -1, 0, -900783, -328762188]\) \(2748251600896/2205\) \(64854011250000\) \([2]\) \(294912\) \(1.9557\) \(\Gamma_0(N)\)-optimal
29400.cc3 29400o2 \([0, -1, 0, -906908, -324058188]\) \(175293437776/4862025\) \(2288049516900000000\) \([2, 2]\) \(589824\) \(2.3023\)  
29400.cc5 29400o3 \([0, -1, 0, 195592, -1062733188]\) \(439608956/259416045\) \(-488320612451280000000\) \([2]\) \(1179648\) \(2.6488\)  
29400.cc2 29400o4 \([0, -1, 0, -2107408, 715574812]\) \(549871953124/200930625\) \(378228593610000000000\) \([2, 2]\) \(1179648\) \(2.6488\)  
29400.cc6 29400o5 \([0, -1, 0, 6467592, 5054524812]\) \(7947184069438/7533176175\) \(-28360660602002400000000\) \([2]\) \(2359296\) \(2.9954\)  
29400.cc1 29400o6 \([0, -1, 0, -29890408, 62893928812]\) \(784478485879202/221484375\) \(833837287500000000000\) \([2]\) \(2359296\) \(2.9954\)