# Properties

 Label 714g Number of curves 6 Conductor 714 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("714.f1")

sage: E.isogeny_class()

## Elliptic curves in class 714g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
714.f5 714g1 [1, 1, 1, -70244, 7127525] [8] 3840 $$\Gamma_0(N)$$-optimal
714.f4 714g2 [1, 1, 1, -90724, 2605541] [2, 4] 7680
714.f2 714g3 [1, 1, 1, -859044, -304722459] [2, 2] 15360
714.f6 714g4 [1, 1, 1, 349916, 20936165] [4] 15360
714.f1 714g5 [1, 1, 1, -13718604, -19563199515] [2] 30720
714.f3 714g6 [1, 1, 1, -292604, -699871003] [2] 30720

## Rank

sage: E.rank()

The elliptic curves in class 714g have rank $$0$$.

## Modular form714.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + q^{8} + q^{9} - 2q^{10} + 4q^{11} - q^{12} - 2q^{13} + q^{14} + 2q^{15} + q^{16} + q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.