Properties

Label 4140.a
Number of curves $4$
Conductor $4140$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4140.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4140.a1 4140e3 \([0, 0, 0, -109488, 13944337]\) \(12444451776495616/912525\) \(10643691600\) \([6]\) \(12096\) \(1.3748\)  
4140.a2 4140e4 \([0, 0, 0, -109263, 14004502]\) \(-772993034343376/6661615005\) \(-1243217238693120\) \([6]\) \(24192\) \(1.7213\)  
4140.a3 4140e1 \([0, 0, 0, -1488, 15037]\) \(31238127616/9703125\) \(113177250000\) \([2]\) \(4032\) \(0.82545\) \(\Gamma_0(N)\)-optimal
4140.a4 4140e2 \([0, 0, 0, 4137, 101662]\) \(41957807024/48205125\) \(-8996233248000\) \([2]\) \(8064\) \(1.1720\)  

Rank

sage: E.rank()
 

The elliptic curves in class 4140.a have rank \(0\).

Complex multiplication

The elliptic curves in class 4140.a do not have complex multiplication.

Modular form 4140.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{5} - 4 q^{7} + 2 q^{13} - 6 q^{17} + 2 q^{19} + O(q^{20})\) 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 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.