# Properties

 Label 672.f Number of curves $4$ Conductor $672$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 672.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
672.f1 672f2 $$[0, 1, 0, -224, -1368]$$ $$2438569736/21$$ $$10752$$ $$$$ $$128$$ $$-0.058633$$
672.f2 672f3 $$[0, 1, 0, -49, 95]$$ $$3241792/567$$ $$2322432$$ $$$$ $$128$$ $$-0.058633$$
672.f3 672f1 $$[0, 1, 0, -14, -24]$$ $$5088448/441$$ $$28224$$ $$[2, 2]$$ $$64$$ $$-0.40521$$ $$\Gamma_0(N)$$-optimal
672.f4 672f4 $$[0, 1, 0, 16, -84]$$ $$830584/7203$$ $$-3687936$$ $$$$ $$128$$ $$-0.058633$$

## Rank

sage: E.rank()

The elliptic curves in class 672.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 672.f do not have complex multiplication.

## Modular form672.2.a.f

sage: E.q_eigenform(10)

$$q + q^{3} - 2 q^{5} - q^{7} + q^{9} - 4 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})$$

## 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. 