# Properties

 Label 291312.be Number of curves $4$ Conductor $291312$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("be1")

sage: E.isogeny_class()

## Elliptic curves in class 291312.be

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
291312.be1 291312be3 $$[0, 0, 0, -211368531, 1182792182546]$$ $$14489843500598257/6246072$$ $$450181859482696384512$$ $$$$ $$42467328$$ $$3.3056$$
291312.be2 291312be4 $$[0, 0, 0, -28258131, -30561915886]$$ $$34623662831857/14438442312$$ $$1040641991967079775895552$$ $$$$ $$42467328$$ $$3.3056$$
291312.be3 291312be2 $$[0, 0, 0, -13276371, 18287610770]$$ $$3590714269297/73410624$$ $$5291026299105271087104$$ $$[2, 2]$$ $$21233664$$ $$2.9590$$
291312.be4 291312be1 $$[0, 0, 0, 40749, 855500690]$$ $$103823/4386816$$ $$-316177108443265499136$$ $$$$ $$10616832$$ $$2.6125$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 291312.be have rank $$1$$.

## Complex multiplication

The elliptic curves in class 291312.be do not have complex multiplication.

## Modular form 291312.2.a.be

sage: E.q_eigenform(10)

$$q - 2q^{5} - q^{7} - 6q^{13} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 