Properties

Label 397800dx
Number of curves $4$
Conductor $397800$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("dx1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 397800dx

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
397800.dx3 397800dx1 \([0, 0, 0, -7050450, -7205635375]\) \(212670222886967296/616241925\) \(112310090831250000\) \([2]\) \(9437184\) \(2.5017\) \(\Gamma_0(N)\)-optimal
397800.dx2 397800dx2 \([0, 0, 0, -7141575, -7009807750]\) \(13813960087661776/714574355625\) \(2083698821002500000000\) \([2, 2]\) \(18874368\) \(2.8483\)  
397800.dx1 397800dx3 \([0, 0, 0, -20304075, 26251829750]\) \(79364416584061444/20404090514925\) \(237993311766085200000000\) \([2]\) \(37748736\) \(3.1949\)  
397800.dx4 397800dx4 \([0, 0, 0, 4562925, -27738477250]\) \(900753985478876/29018422265625\) \(-338470877306250000000000\) \([2]\) \(37748736\) \(3.1949\)  

Rank

sage: E.rank()
 

The elliptic curves in class 397800dx have rank \(1\).

Complex multiplication

The elliptic curves in class 397800dx do not have complex multiplication.

Modular form 397800.2.a.dx

sage: E.q_eigenform(10)
 
\(q + 4q^{7} - q^{13} + q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.