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Properties

Label 12138bb
Number of curves $6$
Conductor $12138$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 12138bb have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(7\)\(1 - T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 12138bb do not have complex multiplication.

Modular form 12138.2.a.bb

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + 2 q^{10} + 4 q^{11} + q^{12} + 6 q^{13} + q^{14} + 2 q^{15} + q^{16} + q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 12138bb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12138.bc5 12138bb1 \([1, 0, 0, -1162, 33572]\) \(-7189057/16128\) \(-389290712832\) \([2]\) \(20480\) \(0.91256\) \(\Gamma_0(N)\)-optimal
12138.bc4 12138bb2 \([1, 0, 0, -24282, 1453140]\) \(65597103937/63504\) \(1532832181776\) \([2, 2]\) \(40960\) \(1.2591\)  
12138.bc3 12138bb3 \([1, 0, 0, -30062, 707520]\) \(124475734657/63011844\) \(1520952732367236\) \([2, 2]\) \(81920\) \(1.6057\)  
12138.bc1 12138bb4 \([1, 0, 0, -388422, 93143592]\) \(268498407453697/252\) \(6082667388\) \([2]\) \(81920\) \(1.6057\)  
12138.bc2 12138bb5 \([1, 0, 0, -264152, -51775458]\) \(84448510979617/933897762\) \(22542021669220578\) \([2]\) \(163840\) \(1.9523\)  
12138.bc6 12138bb6 \([1, 0, 0, 111548, 5493938]\) \(6359387729183/4218578658\) \(-101826233439402402\) \([2]\) \(163840\) \(1.9523\)