# Properties

 Label 116688.h Number of curves $4$ Conductor $116688$ CM no Rank $0$ Graph

# Learn more

Show commands: SageMath
sage: E = EllipticCurve("h1")

sage: E.isogeny_class()

## Elliptic curves in class 116688.h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
116688.h1 116688m3 $$[0, -1, 0, -52723528, 147344310256]$$ $$3957101249824708884951625/772310238681366528$$ $$3163382737638877298688$$ $$[2]$$ $$10948608$$ $$3.1252$$
116688.h2 116688m4 $$[0, -1, 0, -47152968, 179689209840]$$ $$-2830680648734534916567625/1766676274677722124288$$ $$-7236306021079949821083648$$ $$[2]$$ $$21897216$$ $$3.4718$$
116688.h3 116688m1 $$[0, -1, 0, -1605208, -506351120]$$ $$111675519439697265625/37528570137307392$$ $$153717023282411077632$$ $$[2]$$ $$3649536$$ $$2.5759$$ $$\Gamma_0(N)$$-optimal
116688.h4 116688m2 $$[0, -1, 0, 4683432, -3504774672]$$ $$2773679829880629422375/2899504554614368272$$ $$-11876370655700452442112$$ $$[2]$$ $$7299072$$ $$2.9225$$

## Rank

sage: E.rank()

The elliptic curves in class 116688.h have rank $$0$$.

## Complex multiplication

The elliptic curves in class 116688.h do not have complex multiplication.

## Modular form 116688.2.a.h

sage: E.q_eigenform(10)

$$q - q^{3} + 4 q^{7} + q^{9} - q^{11} + q^{13} + q^{17} - 2 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.