Properties

Label 7406.a
Number of curves $6$
Conductor $7406$
CM no
Rank $2$
Graph

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

Elliptic curves in class 7406.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7406.a1 7406d6 \([1, 0, 1, -1444446, 668069456]\) \(2251439055699625/25088\) \(3713924383232\) \([2]\) \(76032\) \(1.9808\)  
7406.a2 7406d5 \([1, 0, 1, -90206, 10450512]\) \(-548347731625/1835008\) \(-271647040602112\) \([2]\) \(38016\) \(1.6343\)  
7406.a3 7406d4 \([1, 0, 1, -18791, 811074]\) \(4956477625/941192\) \(139330194439688\) \([2]\) \(25344\) \(1.4315\)  
7406.a4 7406d2 \([1, 0, 1, -5566, -160170]\) \(128787625/98\) \(14507517122\) \([2]\) \(8448\) \(0.88224\)  
7406.a5 7406d1 \([1, 0, 1, -276, -3586]\) \(-15625/28\) \(-4145004892\) \([2]\) \(4224\) \(0.53566\) \(\Gamma_0(N)\)-optimal
7406.a6 7406d3 \([1, 0, 1, 2369, 74706]\) \(9938375/21952\) \(-3249683835328\) \([2]\) \(12672\) \(1.0850\)  

Rank

sage: E.rank()
 

The elliptic curves in class 7406.a have rank \(2\).

Complex multiplication

The elliptic curves in class 7406.a do not have complex multiplication.

Modular form 7406.2.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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.