# Properties

 Label 1248.f Number of curves $4$ Conductor $1248$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 1248.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1248.f1 1248e3 $$[0, 1, 0, -3744, 86940]$$ $$11339065490696/351$$ $$179712$$ $$$$ $$768$$ $$0.51272$$
1248.f2 1248e2 $$[0, 1, 0, -369, -513]$$ $$1360251712/771147$$ $$3158618112$$ $$$$ $$768$$ $$0.51272$$
1248.f3 1248e1 $$[0, 1, 0, -234, 1296]$$ $$22235451328/123201$$ $$7884864$$ $$[2, 2]$$ $$384$$ $$0.16614$$ $$\Gamma_0(N)$$-optimal
1248.f4 1248e4 $$[0, 1, 0, -104, 2856]$$ $$-245314376/6908733$$ $$-3537271296$$ $$$$ $$768$$ $$0.51272$$

## Rank

sage: E.rank()

The elliptic curves in class 1248.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 1248.f do not have complex multiplication.

## Modular form1248.2.a.f

sage: E.q_eigenform(10)

$$q + q^{3} - 2 q^{5} + q^{9} - 4 q^{11} + q^{13} - 2 q^{15} - 6 q^{17} + 8 q^{19} + O(q^{20})$$

## 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. 