Properties

Label 393008ba
Number of curves $4$
Conductor $393008$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("ba1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 393008ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
393008.ba4 393008ba1 \([0, 0, 0, 1131229, 527829346]\) \(22062729659823/29354283343\) \(-213003892954760900608\) \([2]\) \(10321920\) \(2.5865\) \(\Gamma_0(N)\)-optimal
393008.ba3 393008ba2 \([0, 0, 0, -7009651, 5166502770]\) \(5249244962308257/1448621666569\) \(10511653471226446680064\) \([2, 2]\) \(20643840\) \(2.9330\)  
393008.ba1 393008ba3 \([0, 0, 0, -103296611, 404044863266]\) \(16798320881842096017/2132227789307\) \(15472114051696632737792\) \([2]\) \(41287680\) \(3.2796\)  
393008.ba2 393008ba4 \([0, 0, 0, -40976771, -96836758590]\) \(1048626554636928177/48569076788309\) \(352432464872134575509504\) \([2]\) \(41287680\) \(3.2796\)  

Rank

sage: E.rank()
 

The elliptic curves in class 393008ba have rank \(1\).

Complex multiplication

The elliptic curves in class 393008ba do not have complex multiplication.

Modular form 393008.2.a.ba

sage: E.q_eigenform(10)
 
\(q - 2q^{5} - q^{7} - 3q^{9} - 6q^{13} + 2q^{17} - 8q^{19} + O(q^{20})\)  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.