# Properties

 Label 31200.by Number of curves $4$ Conductor $31200$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 31200.by

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31200.by1 31200bv4 $$[0, 1, 0, -640646033, 6241090296063]$$ $$454357982636417669333824/3003024375$$ $$192193560000000000$$ $$[2]$$ $$5160960$$ $$3.3754$$
31200.by2 31200bv3 $$[0, 1, 0, -42787408, 83359842188]$$ $$1082883335268084577352/251301565117746585$$ $$2010412520941972680000000$$ $$[2]$$ $$5160960$$ $$3.3754$$
31200.by3 31200bv1 $$[0, 1, 0, -40041158, 97503029688]$$ $$7099759044484031233216/577161945398025$$ $$577161945398025000000$$ $$[2, 2]$$ $$2580480$$ $$3.0289$$ $$\Gamma_0(N)$$-optimal
31200.by4 31200bv2 $$[0, 1, 0, -37307408, 111390479688]$$ $$-717825640026599866952/254764560814329735$$ $$-2038116486514637880000000$$ $$[2]$$ $$5160960$$ $$3.3754$$

## Rank

sage: E.rank()

The elliptic curves in class 31200.by have rank $$1$$.

## Complex multiplication

The elliptic curves in class 31200.by do not have complex multiplication.

## Modular form 31200.2.a.by

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.