# Properties

 Label 4410.bi Number of curves $8$ Conductor $4410$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bi1")

sage: E.isogeny_class()

## Elliptic curves in class 4410.bi

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4410.bi1 4410bk7 $$[1, -1, 1, -154893542, -741951322059]$$ $$4791901410190533590281/41160000$$ $$3530133540360000$$ $$[2]$$ $$442368$$ $$3.0245$$
4410.bi2 4410bk6 $$[1, -1, 1, -9681062, -11590632651]$$ $$1169975873419524361/108425318400$$ $$9299218977357926400$$ $$[2, 2]$$ $$221184$$ $$2.6780$$
4410.bi3 4410bk8 $$[1, -1, 1, -8975462, -13352374731]$$ $$-932348627918877961/358766164249920$$ $$-30769982253764512960320$$ $$[2]$$ $$442368$$ $$3.0245$$
4410.bi4 4410bk4 $$[1, -1, 1, -1921667, -1006825809]$$ $$9150443179640281/184570312500$$ $$15829879754882812500$$ $$[2]$$ $$147456$$ $$2.4752$$
4410.bi5 4410bk3 $$[1, -1, 1, -649382, -152913099]$$ $$353108405631241/86318776320$$ $$7403226614433054720$$ $$[4]$$ $$110592$$ $$2.3314$$
4410.bi6 4410bk2 $$[1, -1, 1, -254687, 26034999]$$ $$21302308926361/8930250000$$ $$765912902060250000$$ $$[2, 2]$$ $$73728$$ $$2.1286$$
4410.bi7 4410bk1 $$[1, -1, 1, -219407, 39596631]$$ $$13619385906841/6048000$$ $$518713499808000$$ $$[4]$$ $$36864$$ $$1.7821$$ $$\Gamma_0(N)$$-optimal
4410.bi8 4410bk5 $$[1, -1, 1, 847813, 190527999]$$ $$785793873833639/637994920500$$ $$-54718349548988380500$$ $$[2]$$ $$147456$$ $$2.4752$$

## Rank

sage: E.rank()

The elliptic curves in class 4410.bi have rank $$1$$.

## Complex multiplication

The elliptic curves in class 4410.bi do not have complex multiplication.

## Modular form4410.2.a.bi

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} - 2q^{13} + q^{16} - 6q^{17} - 8q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.