Properties

Label 388080kh
Number of curves $6$
Conductor $388080$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("kh1")
 
E.isogeny_class()
 

Elliptic curves in class 388080kh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
388080.kh5 388080kh1 \([0, 0, 0, -2348589747, -51330783002254]\) \(-4078208988807294650401/880065599546327040\) \(-309165312813579591754595696640\) \([2]\) \(424673280\) \(4.3797\) \(\Gamma_0(N)\)-optimal
388080.kh4 388080kh2 \([0, 0, 0, -39342351027, -3003484724412046]\) \(19170300594578891358373921/671785075055001600\) \(235996774535828885015730585600\) \([2, 2]\) \(849346560\) \(4.7263\)  
388080.kh3 388080kh3 \([0, 0, 0, -41112560307, -2718404203249294]\) \(21876183941534093095979041/3572502915711058560000\) \(1255013242231715633131707432960000\) \([2, 2]\) \(1698693120\) \(5.0728\)  
388080.kh1 388080kh4 \([0, 0, 0, -629472322227, -192226417495801486]\) \(78519570041710065450485106721/96428056919040\) \(33874986588214361472368640\) \([2]\) \(1698693120\) \(5.0728\)  
388080.kh2 388080kh5 \([0, 0, 0, -185139632307, 28064242646581106]\) \(1997773216431678333214187041/187585177195046990066400\) \(65898303508958579714951825139302400\) \([2]\) \(3397386240\) \(5.4194\)  
388080.kh6 388080kh6 \([0, 0, 0, 74591163213, -15255897698663566]\) \(130650216943167617311657439/361816948816603087500000\) \(-127105581924579685218298214400000000\) \([2]\) \(3397386240\) \(5.4194\)  

Rank

sage: E.rank()
 

The elliptic curves in class 388080kh have rank \(1\).

Complex multiplication

The elliptic curves in class 388080kh do not have complex multiplication.

Modular form 388080.2.a.kh

sage: E.q_eigenform(10)
 
\(q + q^{5} - q^{11} + 2 q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.