# Properties

 Label 15680.by Number of curves $4$ Conductor $15680$ CM no Rank $0$ Graph # Learn more

Show commands for: SageMath
sage: E = EllipticCurve("by1")

sage: E.isogeny_class()

## Elliptic curves in class 15680.by

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
15680.by1 15680d3 $$[0, 0, 0, -16268, -773808]$$ $$123505992/4375$$ $$16866160640000$$ $$$$ $$36864$$ $$1.3087$$
15680.by2 15680d2 $$[0, 0, 0, -2548, 32928]$$ $$3796416/1225$$ $$590315622400$$ $$[2, 2]$$ $$18432$$ $$0.96212$$
15680.by3 15680d1 $$[0, 0, 0, -2303, 42532]$$ $$179406144/35$$ $$263533760$$ $$$$ $$9216$$ $$0.61555$$ $$\Gamma_0(N)$$-optimal
15680.by4 15680d4 $$[0, 0, 0, 7252, 225008]$$ $$10941048/12005$$ $$-46280744796160$$ $$$$ $$36864$$ $$1.3087$$

## Rank

sage: E.rank()

The elliptic curves in class 15680.by have rank $$0$$.

## Complex multiplication

The elliptic curves in class 15680.by do not have complex multiplication.

## Modular form 15680.2.a.by

sage: E.q_eigenform(10)

$$q - q^{5} - 3q^{9} + 4q^{11} + 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$ ## 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. 