# Properties

 Label 704a Number of curves 3 Conductor 704 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("704.h1")

sage: E.isogeny_class()

## Elliptic curves in class 704a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
704.h3 704a1 [0, 1, 0, -1, 1] [] 16 $$\Gamma_0(N)$$-optimal
704.h2 704a2 [0, 1, 0, -41, -199] [] 80
704.h1 704a3 [0, 1, 0, -31281, -2139919] [] 400

## Rank

sage: E.rank()

The elliptic curves in class 704a have rank $$1$$.

## Modular form704.2.a.h

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} - 2q^{7} - 2q^{9} - q^{11} - 4q^{13} - q^{15} - 2q^{17} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 