# Properties

 Label 342.c Number of curves $4$ Conductor $342$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 342.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
342.c1 342c3 $$[1, -1, 0, -3852, 92988]$$ $$8671983378625/82308$$ $$60002532$$ $$$$ $$288$$ $$0.65481$$
342.c2 342c4 $$[1, -1, 0, -3762, 97470]$$ $$-8078253774625/846825858$$ $$-617336050482$$ $$$$ $$576$$ $$1.0014$$
342.c3 342c1 $$[1, -1, 0, -72, 0]$$ $$57066625/32832$$ $$23934528$$ $$$$ $$96$$ $$0.10550$$ $$\Gamma_0(N)$$-optimal
342.c4 342c2 $$[1, -1, 0, 288, -216]$$ $$3616805375/2105352$$ $$-1534801608$$ $$$$ $$192$$ $$0.45208$$

## Rank

sage: E.rank()

The elliptic curves in class 342.c have rank $$1$$.

## Complex multiplication

The elliptic curves in class 342.c do not have complex multiplication.

## Modular form342.2.a.c

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - 4q^{7} - q^{8} - 4q^{13} + 4q^{14} + q^{16} - 6q^{17} + q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 