# Properties

 Label 348726.by Number of curves $2$ Conductor $348726$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 348726.by

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
348726.by1 348726by1 $$[1, 1, 1, -1518, 20523]$$ $$56402207875/4451328$$ $$30531658752$$ $$$$ $$307200$$ $$0.75560$$ $$\Gamma_0(N)$$-optimal
348726.by2 348726by2 $$[1, 1, 1, 1522, 95915]$$ $$56844576125/604685088$$ $$-4147535018592$$ $$$$ $$614400$$ $$1.1022$$

## Rank

sage: E.rank()

The elliptic curves in class 348726.by have rank $$1$$.

## Complex multiplication

The elliptic curves in class 348726.by do not have complex multiplication.

## Modular form 348726.2.a.by

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 