# Properties

 Label 12696.c Number of curves $2$ Conductor $12696$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

## Elliptic curves in class 12696.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12696.c1 12696n1 $$[0, -1, 0, -199, 1144]$$ $$4499456/27$$ $$5256144$$ $$$$ $$2304$$ $$0.12899$$ $$\Gamma_0(N)$$-optimal
12696.c2 12696n2 $$[0, -1, 0, -84, 2340]$$ $$-21296/729$$ $$-2270654208$$ $$$$ $$4608$$ $$0.47557$$

## Rank

sage: E.rank()

The elliptic curves in class 12696.c have rank $$1$$.

## Complex multiplication

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

## Modular form 12696.2.a.c

sage: E.q_eigenform(10)

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