# Properties

 Label 12696.s Number of curves $2$ Conductor $12696$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 12696.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12696.s1 12696p2 $$[0, 1, 0, -63069672, -192808566720]$$ $$15043017316604/243$$ $$448184419057161216$$ $$[2]$$ $$1059840$$ $$2.9324$$
12696.s2 12696p1 $$[0, 1, 0, -3938052, -3019719168]$$ $$-14647977776/59049$$ $$-27227203457722543872$$ $$[2]$$ $$529920$$ $$2.5858$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 12696.s have rank $$0$$.

## Complex multiplication

The elliptic curves in class 12696.s do not have complex multiplication.

## Modular form 12696.2.a.s

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.