# Properties

 Label 397800.cf Number of curves $4$ Conductor $397800$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("cf1")

sage: E.isogeny_class()

## Elliptic curves in class 397800.cf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
397800.cf1 397800cf3 $$[0, 0, 0, -7356675, 7680086750]$$ $$1887517194957938/21849165$$ $$509697321120000000$$ $$$$ $$9437184$$ $$2.5489$$
397800.cf2 397800cf2 $$[0, 0, 0, -471675, 113471750]$$ $$994958062276/98903025$$ $$1153604883600000000$$ $$[2, 2]$$ $$4718592$$ $$2.2023$$
397800.cf3 397800cf1 $$[0, 0, 0, -107175, -11551750]$$ $$46689225424/7249905$$ $$21140722980000000$$ $$$$ $$2359296$$ $$1.8557$$ $$\Gamma_0(N)$$-optimal
397800.cf4 397800cf4 $$[0, 0, 0, 581325, 548360750]$$ $$931329171502/6107473125$$ $$-142475133060000000000$$ $$$$ $$9437184$$ $$2.5489$$

## Rank

sage: E.rank()

The elliptic curves in class 397800.cf have rank $$0$$.

## Complex multiplication

The elliptic curves in class 397800.cf do not have complex multiplication.

## Modular form 397800.2.a.cf

sage: E.q_eigenform(10)

$$q - 4q^{11} + q^{13} + q^{17} + 8q^{19} + O(q^{20})$$ ## 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. 