# Properties

 Label 704.c Number of curves $3$ Conductor $704$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("704.c1")

sage: E.isogeny_class()

## Elliptic curves in class 704.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
704.c1 704k3 [0, -1, 0, -31281, 2139919] [] 400
704.c2 704k2 [0, -1, 0, -41, 199] [] 80
704.c3 704k1 [0, -1, 0, -1, -1] [] 16 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form704.2.a.c

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 LMFDB 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 LMFDB labels.