# Properties

 Label 12696.r Number of curves $2$ Conductor $12696$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("r1")

sage: E.isogeny_class()

## Elliptic curves in class 12696.r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12696.r1 12696r2 $$[0, 1, 0, -912, 10080]$$ $$3370318/81$$ $$2018359296$$ $$$$ $$9216$$ $$0.57013$$
12696.r2 12696r1 $$[0, 1, 0, 8, 512]$$ $$4/9$$ $$-112131072$$ $$$$ $$4608$$ $$0.22355$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 12696.2.a.r

sage: E.q_eigenform(10)

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