# Properties

 Label 304e Number of curves $3$ Conductor $304$ CM no Rank $0$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 304e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
304.f3 304e1 $$[0, -1, 0, 11, -3]$$ $$32768/19$$ $$-77824$$ $$[]$$ $$24$$ $$-0.37203$$ $$\Gamma_0(N)$$-optimal
304.f2 304e2 $$[0, -1, 0, -149, 797]$$ $$-89915392/6859$$ $$-28094464$$ $$[]$$ $$72$$ $$0.17728$$
304.f1 304e3 $$[0, -1, 0, -12309, 529757]$$ $$-50357871050752/19$$ $$-77824$$ $$[]$$ $$216$$ $$0.72659$$

## Rank

sage: E.rank()

The elliptic curves in class 304e have rank $$0$$.

## Complex multiplication

The elliptic curves in class 304e do not have complex multiplication.

## Modular form304.2.a.e

sage: E.q_eigenform(10)

$$q + 2q^{3} + 3q^{5} + q^{7} + q^{9} - 3q^{11} - 4q^{13} + 6q^{15} - 3q^{17} - q^{19} + O(q^{20})$$ ## 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 Cremona numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 