# Properties

 Label 234416z Number of curves $2$ Conductor $234416$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 234416z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
234416.z2 234416z1 $$[0, 0, 0, -10780, -429093]$$ $$73598976000/336973$$ $$634312583632$$ $$$$ $$276480$$ $$1.1155$$ $$\Gamma_0(N)$$-optimal
234416.z1 234416z2 $$[0, 0, 0, -16415, 67914]$$ $$16241202000/9332687$$ $$281083210972928$$ $$$$ $$552960$$ $$1.4621$$

## Rank

sage: E.rank()

The elliptic curves in class 234416z have rank $$1$$.

## Complex multiplication

The elliptic curves in class 234416z do not have complex multiplication.

## Modular form 234416.2.a.z

sage: E.q_eigenform(10)

$$q - 3q^{9} + q^{13} - 6q^{17} - 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 