Properties

Label 112.b
Number of curves $4$
Conductor $112$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 112.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
112.b1 112b3 \([0, 0, 0, -299, -1990]\) \(1443468546/7\) \(14336\) \([2]\) \(16\) \(-0.0020328\)  
112.b2 112b4 \([0, 0, 0, -59, 138]\) \(11090466/2401\) \(4917248\) \([4]\) \(16\) \(-0.0020328\)  
112.b3 112b2 \([0, 0, 0, -19, -30]\) \(740772/49\) \(50176\) \([2, 2]\) \(8\) \(-0.34861\)  
112.b4 112b1 \([0, 0, 0, 1, -2]\) \(432/7\) \(-1792\) \([2]\) \(4\) \(-0.69518\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 112.2.a.b

sage: E.q_eigenform(10)
 
\(q + 2q^{5} + q^{7} - 3q^{9} + 4q^{11} + 2q^{13} - 6q^{17} - 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.