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Properties

Label 112.c
Number of curves $6$
Conductor $112$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("c1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 112.c have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(5\) \( 1 + 5 T^{2}\) 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.ag
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 112.c do not have complex multiplication.

Modular form 112.2.a.c

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{3} - q^{7} + q^{9} - 4 q^{13} + 6 q^{17} - 2 q^{19} + O(q^{20})\) 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 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 sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 112.c

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
112.c1 112c6 \([0, -1, 0, -43688, 3529328]\) \(2251439055699625/25088\) \(102760448\) \([2]\) \(144\) \(1.1062\)  
112.c2 112c5 \([0, -1, 0, -2728, 55920]\) \(-548347731625/1835008\) \(-7516192768\) \([2]\) \(72\) \(0.75968\)  
112.c3 112c4 \([0, -1, 0, -568, 4464]\) \(4956477625/941192\) \(3855122432\) \([2]\) \(48\) \(0.55694\)  
112.c4 112c2 \([0, -1, 0, -168, -784]\) \(128787625/98\) \(401408\) \([2]\) \(16\) \(0.0076359\)  
112.c5 112c1 \([0, -1, 0, -8, -16]\) \(-15625/28\) \(-114688\) \([2]\) \(8\) \(-0.33894\) \(\Gamma_0(N)\)-optimal
112.c6 112c3 \([0, -1, 0, 72, 368]\) \(9938375/21952\) \(-89915392\) \([2]\) \(24\) \(0.21037\)