Properties

Label 14a
Number of curves $6$
Conductor $14$
CM no
Rank $0$
Graph

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

Elliptic curves in class 14a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
14.a6 14a1 \([1, 0, 1, 4, -6]\) \(9938375/21952\) \(-21952\) \([6]\) \(1\) \(-0.48278\) \(\Gamma_0(N)\)-optimal
14.a3 14a2 \([1, 0, 1, -36, -70]\) \(4956477625/941192\) \(941192\) \([6]\) \(2\) \(-0.13621\)  
14.a2 14a3 \([1, 0, 1, -171, -874]\) \(-548347731625/1835008\) \(-1835008\) \([2]\) \(3\) \(0.066527\)  
14.a5 14a4 \([1, 0, 1, -1, 0]\) \(-15625/28\) \(-28\) \([6]\) \(3\) \(-1.0321\)  
14.a1 14a5 \([1, 0, 1, -2731, -55146]\) \(2251439055699625/25088\) \(25088\) \([2]\) \(6\) \(0.41310\)  
14.a4 14a6 \([1, 0, 1, -11, 12]\) \(128787625/98\) \(98\) \([6]\) \(6\) \(-0.68551\)  

Rank

sage: E.rank()
 

The elliptic curves in class 14a have rank \(0\).

Complex multiplication

The elliptic curves in class 14a do not have complex multiplication.

Modular form 14.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - 2q^{3} + q^{4} + 2q^{6} + q^{7} - q^{8} + q^{9} - 2q^{12} - 4q^{13} - q^{14} + q^{16} + 6q^{17} - q^{18} + 2q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 3 & 6 & 6 \\ 2 & 1 & 6 & 6 & 3 & 3 \\ 3 & 6 & 1 & 9 & 2 & 18 \\ 3 & 6 & 9 & 1 & 18 & 2 \\ 6 & 3 & 2 & 18 & 1 & 9 \\ 6 & 3 & 18 & 2 & 9 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.