# Properties

 Label 92400hp Number of curves $4$ Conductor $92400$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("hp1")

sage: E.isogeny_class()

## Elliptic curves in class 92400hp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
92400.hk4 92400hp1 $$[0, 1, 0, -1263008, -37752012]$$ $$3481467828171481/2005331497785$$ $$128341215858240000000$$ $$[2]$$ $$2949120$$ $$2.5480$$ $$\Gamma_0(N)$$-optimal
92400.hk2 92400hp2 $$[0, 1, 0, -14385008, -20954220012]$$ $$5143681768032498601/14238434358225$$ $$911259798926400000000$$ $$[2, 2]$$ $$5898240$$ $$2.8946$$
92400.hk3 92400hp3 $$[0, 1, 0, -8715008, -37635360012]$$ $$-1143792273008057401/8897444448004035$$ $$-569436444672258240000000$$ $$[2]$$ $$11796480$$ $$3.2412$$
92400.hk1 92400hp4 $$[0, 1, 0, -230007008, -1342717080012]$$ $$21026497979043461623321/161783881875$$ $$10354168440000000000$$ $$[2]$$ $$11796480$$ $$3.2412$$

## Rank

sage: E.rank()

The elliptic curves in class 92400hp have rank $$1$$.

## Complex multiplication

The elliptic curves in class 92400hp do not have complex multiplication.

## Modular form 92400.2.a.hp

sage: E.q_eigenform(10)

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

## 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 Cremona 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 Cremona labels.