# Properties

 Label 234.e Number of curves $3$ Conductor $234$ CM no Rank $0$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 234.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
234.e1 234e3 $$[1, -1, 1, -4136, 103403]$$ $$-10730978619193/6656$$ $$-4852224$$ $$[3]$$ $$180$$ $$0.60369$$
234.e2 234e2 $$[1, -1, 1, -41, 209]$$ $$-10218313/17576$$ $$-12812904$$ $$[3]$$ $$60$$ $$0.054386$$
234.e3 234e1 $$[1, -1, 1, 4, -7]$$ $$12167/26$$ $$-18954$$ $$[]$$ $$20$$ $$-0.49492$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 234.e have rank $$0$$.

## Complex multiplication

The elliptic curves in class 234.e do not have complex multiplication.

## Modular form234.2.a.e

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.