# Properties

 Label 1638.l Number of curves $3$ Conductor $1638$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("l1")

E.isogeny_class()

## Elliptic curves in class 1638.l

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.l1 1638t3 $$[1, -1, 1, -234509, 43769909]$$ $$-1956469094246217097/36641439744$$ $$-26711609573376$$ $$[3]$$ $$15552$$ $$1.7008$$
1638.l2 1638t2 $$[1, -1, 1, -1094, 133589]$$ $$-198461344537/10417365504$$ $$-7594259452416$$ $$[3]$$ $$5184$$ $$1.1515$$
1638.l3 1638t1 $$[1, -1, 1, 121, -4921]$$ $$270840023/14329224$$ $$-10446004296$$ $$[]$$ $$1728$$ $$0.60215$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1638.l have rank $$1$$.

## Complex multiplication

The elliptic curves in class 1638.l do not have complex multiplication.

## Modular form1638.2.a.l

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - 3 q^{5} + q^{7} + q^{8} - 3 q^{10} - 3 q^{11} + q^{13} + q^{14} + q^{16} + 3 q^{17} - 7 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.