# Properties

 Label 1248.a Number of curves $2$ Conductor $1248$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 1248.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1248.a1 1248a1 $$[0, -1, 0, -14, -12]$$ $$5088448/1053$$ $$67392$$ $$$$ $$128$$ $$-0.35830$$ $$\Gamma_0(N)$$-optimal
1248.a2 1248a2 $$[0, -1, 0, 31, -111]$$ $$778688/1521$$ $$-6230016$$ $$$$ $$256$$ $$-0.011725$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1248.2.a.a

sage: E.q_eigenform(10)

$$q - q^{3} - 2 q^{5} - 2 q^{7} + q^{9} + 6 q^{11} - q^{13} + 2 q^{15} - 2 q^{17} + 6 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 