Properties

Label 411840ba
Number of curves $4$
Conductor $411840$
CM no
Rank $0$
Graph

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

Elliptic curves in class 411840ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
411840.ba3 411840ba1 \([0, 0, 0, -38028, 2064112]\) \(31824875809/8785920\) \(1679015458897920\) \([2]\) \(1769472\) \(1.6289\) \(\Gamma_0(N)\)-optimal
411840.ba2 411840ba2 \([0, 0, 0, -222348, -38707472]\) \(6361447449889/294465600\) \(56273252489625600\) \([2, 2]\) \(3538944\) \(1.9755\)  
411840.ba4 411840ba3 \([0, 0, 0, 123252, -148055312]\) \(1083523132511/50179392120\) \(-9589431168002949120\) \([2]\) \(7077888\) \(2.3220\)  
411840.ba1 411840ba4 \([0, 0, 0, -3517068, -2538741008]\) \(25176685646263969/57915000\) \(11067728855040000\) \([2]\) \(7077888\) \(2.3220\)  

Rank

sage: E.rank()
 

The elliptic curves in class 411840ba have rank \(0\).

Complex multiplication

The elliptic curves in class 411840ba do not have complex multiplication.

Modular form 411840.2.a.ba

sage: E.q_eigenform(10)
 
\(q - q^{5} - 4 q^{7} + q^{11} + q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.