Properties

Label 41070.d
Number of curves $6$
Conductor $41070$
CM no
Rank $1$
Graph

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

Elliptic curves in class 41070.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
41070.d1 41070b4 \([1, 1, 0, -466835873, 3882153327477]\) \(4385367890843575421521/24975000000\) \(64079017064775000000\) \([2]\) \(9455616\) \(3.4123\)  
41070.d2 41070b6 \([1, 1, 0, -414978153, -3239790411267]\) \(3080272010107543650001/15465841417699560\) \(39681117762797761119680040\) \([2]\) \(18911232\) \(3.7589\)  
41070.d3 41070b3 \([1, 1, 0, -40145953, 10979326453]\) \(2788936974993502801/1593609593601600\) \(4088766219939382544654400\) \([2, 2]\) \(9455616\) \(3.4123\)  
41070.d4 41070b2 \([1, 1, 0, -29193953, 60576553653]\) \(1072487167529950801/2554882560000\) \(6555129656085527040000\) \([2, 2]\) \(4727808\) \(3.0657\)  
41070.d5 41070b1 \([1, 1, 0, -1156833, 1648134837]\) \(-66730743078481/419010969600\) \(-1075067510363416166400\) \([2]\) \(2363904\) \(2.7191\) \(\Gamma_0(N)\)-optimal
41070.d6 41070b5 \([1, 1, 0, 159454247, 87745563373]\) \(174751791402194852399/102423900876336360\) \(-262791707391214442018931240\) \([2]\) \(18911232\) \(3.7589\)  

Rank

sage: E.rank()
 

The elliptic curves in class 41070.d have rank \(1\).

Complex multiplication

The elliptic curves in class 41070.d do not have complex multiplication.

Modular form 41070.2.a.d

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

Isogeny matrix

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.