Properties

Label 481338.ba
Number of curves $4$
Conductor $481338$
CM no
Rank $0$
Graph

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

Elliptic curves in class 481338.ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
481338.ba1 481338ba3 \([1, -1, 0, -3588495147, -82725366978171]\) \(3957101249824708884951625/772310238681366528\) \(997413935387729668060741632\) \([2]\) \(437944320\) \(4.1803\)  
481338.ba2 481338ba4 \([1, -1, 0, -3209348907, -100888670922363]\) \(-2830680648734534916567625/1766676274677722124288\) \(-2281605820338523921400588931072\) \([2]\) \(875888640\) \(4.5269\)  
481338.ba3 481338ba1 \([1, -1, 0, -109254492, 284651829072]\) \(111675519439697265625/37528570137307392\) \(48466946254702428674926848\) \([2]\) \(145981440\) \(3.6310\) \(\Gamma_0(N)\)-optimal*
481338.ba4 481338ba2 \([1, -1, 0, 318766068, 1967029442208]\) \(2773679829880629422375/2899504554614368272\) \(-3744617258254067770457879568\) \([2]\) \(291962880\) \(3.9776\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 481338.ba1.

Rank

sage: E.rank()
 

The elliptic curves in class 481338.ba have rank \(0\).

Complex multiplication

The elliptic curves in class 481338.ba do not have complex multiplication.

Modular form 481338.2.a.ba

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + 4 q^{7} - q^{8} - q^{13} - 4 q^{14} + q^{16} + q^{17} - 2 q^{19} + O(q^{20})\)  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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.