# Properties

 Label 369600.hl Number of curves $6$ Conductor $369600$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("hl1")

sage: E.isogeny_class()

## Elliptic curves in class 369600.hl

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
369600.hl1 369600hl5 $$[0, -1, 0, -4878720033, 131163302591937]$$ $$3135316978843283198764801/571725$$ $$2341785600000000$$ $$[2]$$ $$94371840$$ $$3.7436$$
369600.hl2 369600hl3 $$[0, -1, 0, -304920033, 2049502391937]$$ $$765458482133960722801/326869475625$$ $$1338857372160000000000$$ $$[2, 2]$$ $$47185920$$ $$3.3970$$
369600.hl3 369600hl6 $$[0, -1, 0, -303408033, 2070832175937]$$ $$-754127868744065783521/15825714261328125$$ $$-64822125614400000000000000$$ $$[2]$$ $$94371840$$ $$3.7436$$
369600.hl4 369600hl4 $$[0, -1, 0, -40712033, -52712856063]$$ $$1821931919215868881/761147600816295$$ $$3117660572943544320000000$$ $$[2]$$ $$47185920$$ $$3.3970$$
369600.hl5 369600hl2 $$[0, -1, 0, -19152033, 31694543937]$$ $$189674274234120481/3859869269025$$ $$15810024525926400000000$$ $$[2, 2]$$ $$23592960$$ $$3.0504$$
369600.hl6 369600hl1 $$[0, -1, 0, 55967, 1480359937]$$ $$4733169839/231139696095$$ $$-946748195205120000000$$ $$[2]$$ $$11796480$$ $$2.7039$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 369600.hl have rank $$1$$.

## Complex multiplication

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

## Modular form 369600.2.a.hl

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.