# Properties

 Label 12696.p Number of curves $2$ Conductor $12696$ CM no Rank $2$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 12696.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12696.p1 12696s2 $$[0, 1, 0, -119224, 15805376]$$ $$15043017316604/243$$ $$3027538944$$ $$$$ $$46080$$ $$1.3646$$
12696.p2 12696s1 $$[0, 1, 0, -7444, 245600]$$ $$-14647977776/59049$$ $$-183922990848$$ $$$$ $$23040$$ $$1.0181$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 12696.p have rank $$2$$.

## Complex multiplication

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

## Modular form 12696.2.a.p

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. 