# Properties

 Label 111090bd Number of curves $4$ Conductor $111090$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bd1")

sage: E.isogeny_class()

## Elliptic curves in class 111090bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
111090.bc3 111090bd1 $$[1, 0, 1, -67248343, -227341860742]$$ $$-227196402372228188089/19338934824115200$$ $$-2862856409000952270412800$$ $$[2]$$ $$24330240$$ $$3.4377$$ $$\Gamma_0(N)$$-optimal
111090.bc2 111090bd2 $$[1, 0, 1, -1097063223, -13986080583494]$$ $$986396822567235411402169/6336721794060000$$ $$938062244129347019340000$$ $$[2]$$ $$48660480$$ $$3.7842$$
111090.bc4 111090bd3 $$[1, 0, 1, 398686922, -8372434744]$$ $$47342661265381757089751/27397579603968000000$$ $$-4055825053121670807552000000$$ $$[2]$$ $$72990720$$ $$3.9870$$
111090.bc1 111090bd4 $$[1, 0, 1, -1594754358, -67378296632]$$ $$3029968325354577848895529/1753440696000000000000$$ $$259572152241138744000000000000$$ $$[2]$$ $$145981440$$ $$4.3335$$

## Rank

sage: E.rank()

The elliptic curves in class 111090bd have rank $$0$$.

## Complex multiplication

The elliptic curves in class 111090bd do not have complex multiplication.

## Modular form 111090.2.a.bd

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + q^{12} - 4q^{13} + q^{14} + q^{15} + q^{16} - q^{18} - 2q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.