# Properties

 Label 37440ck Number of curves $4$ Conductor $37440$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("ck1")

sage: E.isogeny_class()

## Elliptic curves in class 37440ck

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
37440.fu4 37440ck1 $$[0, 0, 0, -732, -27056]$$ $$-3631696/24375$$ $$-291133440000$$ $$$$ $$49152$$ $$0.88311$$ $$\Gamma_0(N)$$-optimal
37440.fu3 37440ck2 $$[0, 0, 0, -18732, -984656]$$ $$15214885924/38025$$ $$1816672665600$$ $$[2, 2]$$ $$98304$$ $$1.2297$$
37440.fu2 37440ck3 $$[0, 0, 0, -25932, -158096]$$ $$20183398562/11567205$$ $$1105263649751040$$ $$$$ $$196608$$ $$1.5763$$
37440.fu1 37440ck4 $$[0, 0, 0, -299532, -63097616]$$ $$31103978031362/195$$ $$18632540160$$ $$$$ $$196608$$ $$1.5763$$

## Rank

sage: E.rank()

The elliptic curves in class 37440ck have rank $$1$$.

## Complex multiplication

The elliptic curves in class 37440ck do not have complex multiplication.

## Modular form 37440.2.a.ck

sage: E.q_eigenform(10)

$$q + q^{5} + 4q^{7} + 4q^{11} - q^{13} - 6q^{17} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 