# Properties

 Label 346560ef Number of curves $4$ Conductor $346560$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 346560ef

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
346560.ef3 346560ef1 $$[0, -1, 0, -716705, -231902175]$$ $$3301293169/22800$$ $$281187735778099200$$ $$[2]$$ $$4423680$$ $$2.1817$$ $$\Gamma_0(N)$$-optimal
346560.ef2 346560ef2 $$[0, -1, 0, -1178785, 104214817]$$ $$14688124849/8122500$$ $$100173130870947840000$$ $$[2, 2]$$ $$8847360$$ $$2.5283$$
346560.ef1 346560ef3 $$[0, -1, 0, -14348065, 20893240225]$$ $$26487576322129/44531250$$ $$549194796441600000000$$ $$[2]$$ $$17694720$$ $$2.8749$$
346560.ef4 346560ef4 $$[0, -1, 0, 4597215, 819283617]$$ $$871257511151/527800050$$ $$-6509250043994190643200$$ $$[2]$$ $$17694720$$ $$2.8749$$

## Rank

sage: E.rank()

The elliptic curves in class 346560ef have rank $$1$$.

## Complex multiplication

The elliptic curves in class 346560ef do not have complex multiplication.

## Modular form 346560.2.a.ef

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.