# Properties

 Label 288.b Number of curves $4$ Conductor $288$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("b1")

E.isogeny_class()

## Elliptic curves in class 288.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
288.b1 288b2 $$[0, 0, 0, -291, -1910]$$ $$7301384/3$$ $$1119744$$ $$[2]$$ $$64$$ $$0.12309$$
288.b2 288b3 $$[0, 0, 0, -156, 736]$$ $$140608/3$$ $$8957952$$ $$[4]$$ $$64$$ $$0.12309$$
288.b3 288b1 $$[0, 0, 0, -21, -20]$$ $$21952/9$$ $$419904$$ $$[2, 2]$$ $$32$$ $$-0.22349$$ $$\Gamma_0(N)$$-optimal
288.b4 288b4 $$[0, 0, 0, 69, -146]$$ $$97336/81$$ $$-30233088$$ $$[2]$$ $$64$$ $$0.12309$$

## Rank

sage: E.rank()

The elliptic curves in class 288.b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 288.b do not have complex multiplication.

## Modular form288.2.a.b

sage: E.q_eigenform(10)

$$q - 2 q^{5} - 4 q^{7} - 4 q^{11} - 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.