# Properties

 Label 784.b Number of curves $6$ Conductor $784$ CM no Rank $1$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 784.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
784.b1 784j6 $$[0, 1, 0, -2140728, -1206278060]$$ $$2251439055699625/25088$$ $$12089663946752$$ $$[2]$$ $$6912$$ $$2.0792$$
784.b2 784j5 $$[0, 1, 0, -133688, -18913196]$$ $$-548347731625/1835008$$ $$-884272562962432$$ $$[2]$$ $$3456$$ $$1.7326$$
784.b3 784j4 $$[0, 1, 0, -27848, -1475468]$$ $$4956477625/941192$$ $$453551299002368$$ $$[2]$$ $$2304$$ $$1.5299$$
784.b4 784j2 $$[0, 1, 0, -8248, 285396]$$ $$128787625/98$$ $$47225249792$$ $$[2]$$ $$768$$ $$0.98059$$
784.b5 784j1 $$[0, 1, 0, -408, 6292]$$ $$-15625/28$$ $$-13492928512$$ $$[2]$$ $$384$$ $$0.63402$$ $$\Gamma_0(N)$$-optimal
784.b6 784j3 $$[0, 1, 0, 3512, -133260]$$ $$9938375/21952$$ $$-10578455953408$$ $$[2]$$ $$1152$$ $$1.1833$$

## Rank

sage: E.rank()

The elliptic curves in class 784.b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 784.b do not have complex multiplication.

## Modular form784.2.a.b

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.