# Properties

 Label 466752.ef Number of curves $2$ Conductor $466752$ CM no Rank $1$ Graph

# Learn more

Show commands: SageMath
sage: E = EllipticCurve("ef1")

sage: E.isogeny_class()

## Elliptic curves in class 466752.ef

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
466752.ef1 466752ef1 $$[0, 1, 0, -2525281, -1323593953]$$ $$6793805286030262681/1048227429629952$$ $$274786531312914137088$$ $$[2]$$ $$28901376$$ $$2.6453$$ $$\Gamma_0(N)$$-optimal
466752.ef2 466752ef2 $$[0, 1, 0, 4396959, -7297487073]$$ $$35862531227445945959/108547797844556928$$ $$-28455153918163531333632$$ $$[2]$$ $$57802752$$ $$2.9919$$

## Rank

sage: E.rank()

The elliptic curves in class 466752.ef have rank $$1$$.

## Complex multiplication

The elliptic curves in class 466752.ef do not have complex multiplication.

## Modular form 466752.2.a.ef

sage: E.q_eigenform(10)

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

## 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 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.