# Properties

 Label 54720bc Number of curves $4$ Conductor $54720$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bc1")

sage: E.isogeny_class()

## Elliptic curves in class 54720bc

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54720.bj3 54720bc1 $$[0, 0, 0, -17868, -913808]$$ $$3301293169/22800$$ $$4357147852800$$ $$$$ $$98304$$ $$1.2588$$ $$\Gamma_0(N)$$-optimal
54720.bj2 54720bc2 $$[0, 0, 0, -29388, 408688]$$ $$14688124849/8122500$$ $$1552233922560000$$ $$[2, 2]$$ $$196608$$ $$1.6054$$
54720.bj4 54720bc3 $$[0, 0, 0, 114612, 3231088]$$ $$871257511151/527800050$$ $$-100864160287948800$$ $$$$ $$393216$$ $$1.9519$$
54720.bj1 54720bc4 $$[0, 0, 0, -357708, 82226032]$$ $$26487576322129/44531250$$ $$8510054400000000$$ $$$$ $$393216$$ $$1.9519$$

## Rank

sage: E.rank()

The elliptic curves in class 54720bc have rank $$1$$.

## Complex multiplication

The elliptic curves in class 54720bc do not have complex multiplication.

## Modular form 54720.2.a.bc

sage: E.q_eigenform(10)

$$q - q^{5} + 4q^{11} - 2q^{13} - 2q^{17} + q^{19} + O(q^{20})$$ ## 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 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. 