# Properties

 Label 3696.s Number of curves $4$ Conductor $3696$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("s1")

sage: E.isogeny_class()

## Elliptic curves in class 3696.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3696.s1 3696u4 $$[0, 1, 0, -3624, 82740]$$ $$1285429208617/614922$$ $$2518720512$$ $$$$ $$3072$$ $$0.75881$$
3696.s2 3696u3 $$[0, 1, 0, -2024, -35148]$$ $$223980311017/4278582$$ $$17525071872$$ $$$$ $$3072$$ $$0.75881$$
3696.s3 3696u2 $$[0, 1, 0, -264, 756]$$ $$498677257/213444$$ $$874266624$$ $$[2, 2]$$ $$1536$$ $$0.41223$$
3696.s4 3696u1 $$[0, 1, 0, 56, 116]$$ $$4657463/3696$$ $$-15138816$$ $$$$ $$768$$ $$0.065661$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 3696.s have rank $$1$$.

## Complex multiplication

The elliptic curves in class 3696.s do not have complex multiplication.

## Modular form3696.2.a.s

sage: E.q_eigenform(10)

$$q + q^{3} - 2 q^{5} - q^{7} + q^{9} - q^{11} + 2 q^{13} - 2 q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 