# Properties

 Label 78400hn Number of curves $3$ Conductor $78400$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("hn1")

sage: E.isogeny_class()

## Elliptic curves in class 78400hn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
78400.hf2 78400hn1 $$[0, 1, 0, -6533, 223313]$$ $$-262144/35$$ $$-4117715000000$$ $$[]$$ $$110592$$ $$1.1531$$ $$\Gamma_0(N)$$-optimal
78400.hf3 78400hn2 $$[0, 1, 0, 42467, -560687]$$ $$71991296/42875$$ $$-5044200875000000$$ $$[]$$ $$331776$$ $$1.7024$$
78400.hf1 78400hn3 $$[0, 1, 0, -643533, -208075687]$$ $$-250523582464/13671875$$ $$-1608482421875000000$$ $$[]$$ $$995328$$ $$2.2517$$

## Rank

sage: E.rank()

The elliptic curves in class 78400hn have rank $$1$$.

## Complex multiplication

The elliptic curves in class 78400hn do not have complex multiplication.

## Modular form 78400.2.a.hn

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{9} - 3q^{11} - 5q^{13} + 3q^{17} - 2q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.