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Properties

Label 45486p
Number of curves $6$
Conductor $45486$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve([1, -1, 0, -1692, 54652]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 45486p have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(7\)\(1 + T\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - T + 5 T^{2}\) 1.5.ab
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + T + 13 T^{2}\) 1.13.b
\(17\) \( 1 + 4 T + 17 T^{2}\) 1.17.e
\(23\) \( 1 + T + 23 T^{2}\) 1.23.b
\(29\) \( 1 - 8 T + 29 T^{2}\) 1.29.ai
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 45486p do not have complex multiplication.

Modular form 45486.2.a.p

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{7} - q^{8} + 4 q^{13} - q^{14} + q^{16} - 6 q^{17} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 45486p

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
45486.m5 45486p1 \([1, -1, 0, -1692, 54652]\) \(-15625/28\) \(-960300522972\) \([2]\) \(57024\) \(0.98944\) \(\Gamma_0(N)\)-optimal
45486.m4 45486p2 \([1, -1, 0, -34182, 2439418]\) \(128787625/98\) \(3361051830402\) \([2]\) \(114048\) \(1.3360\)  
45486.m6 45486p3 \([1, -1, 0, 14553, -1137731]\) \(9938375/21952\) \(-752875610010048\) \([2]\) \(171072\) \(1.5387\)  
45486.m3 45486p4 \([1, -1, 0, -115407, -12340283]\) \(4956477625/941192\) \(32279541779180808\) \([2]\) \(342144\) \(1.8853\)  
45486.m2 45486p5 \([1, -1, 0, -554022, -159042380]\) \(-548347731625/1835008\) \(-62934255073492992\) \([2]\) \(513216\) \(2.0881\)  
45486.m1 45486p6 \([1, -1, 0, -8871462, -10168249676]\) \(2251439055699625/25088\) \(860429268582912\) \([2]\) \(1026432\) \(2.4346\)