# Properties

 Label 392a Number of curves $4$ Conductor $392$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 392a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
392.d4 392a1 [0, 0, 0, 49, -686]  96 $$\Gamma_0(N)$$-optimal
392.d3 392a2 [0, 0, 0, -931, -10290] [2, 2] 192
392.d1 392a3 [0, 0, 0, -14651, -682570]  384
392.d2 392a4 [0, 0, 0, -2891, 47334]  384

## Rank

sage: E.rank()

The elliptic curves in class 392a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 392a do not have complex multiplication.

## Modular form392.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{5} - 3q^{9} - 4q^{11} - 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 