# Properties

 Label 369600.pg Number of curves $6$ Conductor $369600$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("pg1")

sage: E.isogeny_class()

## Elliptic curves in class 369600.pg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
369600.pg1 369600pg5 $$[0, 1, 0, -21200033, 37562148063]$$ $$257260669489908001/14267882475$$ $$58441246617600000000$$ $$$$ $$25165824$$ $$2.8583$$
369600.pg2 369600pg3 $$[0, 1, 0, -1400033, 516348063]$$ $$74093292126001/14707625625$$ $$60242434560000000000$$ $$[2, 2]$$ $$12582912$$ $$2.5117$$
369600.pg3 369600pg2 $$[0, 1, 0, -432033, -102203937]$$ $$2177286259681/161417025$$ $$661164134400000000$$ $$[2, 2]$$ $$6291456$$ $$2.1652$$
369600.pg4 369600pg1 $$[0, 1, 0, -424033, -106419937]$$ $$2058561081361/12705$$ $$52039680000000$$ $$$$ $$3145728$$ $$1.8186$$ $$\Gamma_0(N)$$-optimal
369600.pg5 369600pg4 $$[0, 1, 0, 407967, -450803937]$$ $$1833318007919/22507682505$$ $$-92191467540480000000$$ $$$$ $$12582912$$ $$2.5117$$
369600.pg6 369600pg6 $$[0, 1, 0, 2911967, 3073364063]$$ $$666688497209279/1381398046875$$ $$-5658206400000000000000$$ $$$$ $$25165824$$ $$2.8583$$

## Rank

sage: E.rank()

The elliptic curves in class 369600.pg have rank $$0$$.

## Complex multiplication

The elliptic curves in class 369600.pg do not have complex multiplication.

## Modular form 369600.2.a.pg

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 