# Properties

 Label 114.b Number of curves $4$ Conductor $114$ CM no Rank $0$ Graph # Learn more

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

sage: E.isogeny_class()

## Elliptic curves in class 114.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
114.b1 114c3 $$[1, 1, 1, -87552, -10007679]$$ $$74220219816682217473/16416$$ $$16416$$ $$$$ $$240$$ $$1.1014$$
114.b2 114c2 $$[1, 1, 1, -5472, -158079]$$ $$18120364883707393/269485056$$ $$269485056$$ $$[2, 2]$$ $$120$$ $$0.75487$$
114.b3 114c4 $$[1, 1, 1, -5312, -167551]$$ $$-16576888679672833/2216253521952$$ $$-2216253521952$$ $$$$ $$240$$ $$1.1014$$
114.b4 114c1 $$[1, 1, 1, -352, -2431]$$ $$4824238966273/537919488$$ $$537919488$$ $$$$ $$60$$ $$0.40829$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 114.b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 114.b do not have complex multiplication.

## Modular form114.2.a.b

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} + q^{8} + q^{9} + 2q^{10} - 4q^{11} - q^{12} + 2q^{13} - 2q^{15} + q^{16} - 6q^{17} + q^{18} - q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 