# Properties

 Label 456.d Number of curves $4$ Conductor $456$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("d1")

E.isogeny_class()

## Elliptic curves in class 456.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
456.d1 456b3 $$[0, 1, 0, -1272, 16560]$$ $$111223479026/3518667$$ $$7206230016$$ $$[2]$$ $$192$$ $$0.66620$$
456.d2 456b2 $$[0, 1, 0, -192, -720]$$ $$768400132/263169$$ $$269485056$$ $$[2, 2]$$ $$96$$ $$0.31963$$
456.d3 456b1 $$[0, 1, 0, -172, -928]$$ $$2211014608/513$$ $$131328$$ $$[2]$$ $$48$$ $$-0.026945$$ $$\Gamma_0(N)$$-optimal
456.d4 456b4 $$[0, 1, 0, 568, -4368]$$ $$9878111854/10097379$$ $$-20679432192$$ $$[2]$$ $$192$$ $$0.66620$$

## Rank

sage: E.rank()

The elliptic curves in class 456.d have rank $$0$$.

## Complex multiplication

The elliptic curves in class 456.d do not have complex multiplication.

## Modular form456.2.a.d

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.