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Properties

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

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, -1444446, 668069456]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -1444446, 668069456]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -1444446, 668069456]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

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

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(7\)\(1 + T\)
\(23\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 2 T + 3 T^{2}\) 1.3.c
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 7406.2.a.a

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} - q^{7} - q^{8} + q^{9} - 2 q^{12} - 4 q^{13} + q^{14} + q^{16} - 6 q^{17} - q^{18} - 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 7406.a

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
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 \([1, 0, 1, 2369, 74706]\) \(9938375/21952\) \(-3249683835328\) \([2]\) \(12672\) \(1.0850\)