# Properties

 Label 102.b Number of curves $4$ Conductor $102$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 102.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
102.b1 102c3 $$[1, 0, 1, -751, -6046]$$ $$46753267515625/11591221248$$ $$11591221248$$ $$$$ $$72$$ $$0.64143$$
102.b2 102c1 $$[1, 0, 1, -256, 1550]$$ $$1845026709625/793152$$ $$793152$$ $$$$ $$24$$ $$0.092120$$ $$\Gamma_0(N)$$-optimal
102.b3 102c2 $$[1, 0, 1, -216, 2062]$$ $$-1107111813625/1228691592$$ $$-1228691592$$ $$$$ $$48$$ $$0.43869$$
102.b4 102c4 $$[1, 0, 1, 1809, -37790]$$ $$655215969476375/1001033261568$$ $$-1001033261568$$ $$$$ $$144$$ $$0.98800$$

## Rank

sage: E.rank()

The elliptic curves in class 102.b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 102.b do not have complex multiplication.

## Modular form102.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{6} + 2 q^{7} - q^{8} + q^{9} + q^{12} + 2 q^{13} - 2 q^{14} + q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 