# Properties

 Label 7406.i Number of curves $2$ Conductor $7406$ CM no Rank $1$ Graph

# Learn more

Show commands: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 7406.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7406.i1 7406g2 $$[1, 1, 1, -7417, 203509]$$ $$304821217/51842$$ $$7674476557538$$ $$[2]$$ $$25344$$ $$1.1931$$
7406.i2 7406g1 $$[1, 1, 1, -2127, -35599]$$ $$7189057/644$$ $$95335112516$$ $$[2]$$ $$12672$$ $$0.84648$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 7406.i have rank $$1$$.

## Complex multiplication

The elliptic curves in class 7406.i do not have complex multiplication.

## Modular form7406.2.a.i

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.