# Properties

 Label 119952gp Number of curves $4$ Conductor $119952$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("gp1")

sage: E.isogeny_class()

## Elliptic curves in class 119952gp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
119952.bt4 119952gp1 $$[0, 0, 0, 6909, -59726590]$$ $$103823/4386816$$ $$-1541079825861574656$$ $$$$ $$1769472$$ $$2.1688$$ $$\Gamma_0(N)$$-optimal
119952.bt3 119952gp2 $$[0, 0, 0, -2251011, -1276745470]$$ $$3590714269297/73410624$$ $$25789007710902288384$$ $$[2, 2]$$ $$3538944$$ $$2.5154$$
119952.bt2 119952gp3 $$[0, 0, 0, -4791171, 2133673346]$$ $$34623662831857/14438442312$$ $$5072196363806768136192$$ $$$$ $$7077888$$ $$2.8620$$
119952.bt1 119952gp4 $$[0, 0, 0, -35837571, -82576372606]$$ $$14489843500598257/6246072$$ $$2194232798931812352$$ $$$$ $$7077888$$ $$2.8620$$

## Rank

sage: E.rank()

The elliptic curves in class 119952gp have rank $$1$$.

## Complex multiplication

The elliptic curves in class 119952gp do not have complex multiplication.

## Modular form 119952.2.a.gp

sage: E.q_eigenform(10)

$$q - 2q^{5} + 6q^{13} + q^{17} + O(q^{20})$$ ## 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. 