# Properties

 Label 412224b Number of curves $2$ Conductor $412224$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 412224b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
412224.b1 412224b1 $$[0, -1, 0, -31672545, 40482638721]$$ $$13403946614821979039929/5057590268826067968$$ $$1325816943431140761403392$$ $$$$ $$121405440$$ $$3.3282$$ $$\Gamma_0(N)$$-optimal
412224.b2 412224b2 $$[0, -1, 0, 99082015, 288785548161]$$ $$410363075617640914325831/374944243169850027552$$ $$-98289383681517165622591488$$ $$$$ $$242810880$$ $$3.6748$$

## Rank

sage: E.rank()

The elliptic curves in class 412224b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 412224b do not have complex multiplication.

## Modular form 412224.2.a.b

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 