# Properties

 Label 481338.ba Number of curves $4$ Conductor $481338$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("ba1")

sage: E.isogeny_class()

## Elliptic curves in class 481338.ba

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
481338.ba1 481338ba3 $$[1, -1, 0, -3588495147, -82725366978171]$$ $$3957101249824708884951625/772310238681366528$$ $$997413935387729668060741632$$ $$[2]$$ $$437944320$$ $$4.1803$$
481338.ba2 481338ba4 $$[1, -1, 0, -3209348907, -100888670922363]$$ $$-2830680648734534916567625/1766676274677722124288$$ $$-2281605820338523921400588931072$$ $$[2]$$ $$875888640$$ $$4.5269$$
481338.ba3 481338ba1 $$[1, -1, 0, -109254492, 284651829072]$$ $$111675519439697265625/37528570137307392$$ $$48466946254702428674926848$$ $$[2]$$ $$145981440$$ $$3.6310$$ $$\Gamma_0(N)$$-optimal*
481338.ba4 481338ba2 $$[1, -1, 0, 318766068, 1967029442208]$$ $$2773679829880629422375/2899504554614368272$$ $$-3744617258254067770457879568$$ $$[2]$$ $$291962880$$ $$3.9776$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 481338.ba1.

## Rank

sage: E.rank()

The elliptic curves in class 481338.ba have rank $$0$$.

## Complex multiplication

The elliptic curves in class 481338.ba do not have complex multiplication.

## Modular form 481338.2.a.ba

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.