# Properties

 Label 133848z Number of curves 4 Conductor 133848 CM no Rank 2 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("133848.g1")

sage: E.isogeny_class()

## Elliptic curves in class 133848z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
133848.g4 133848z1 [0, 0, 0, 1014, -340535]  368640 $$\Gamma_0(N)$$-optimal
133848.g3 133848z2 [0, 0, 0, -67431, -6569030] [2, 2] 737280
133848.g2 133848z3 [0, 0, 0, -158691, 15023086]  1474560
133848.g1 133848z4 [0, 0, 0, -1071291, -426784826]  1474560

## Rank

sage: E.rank()

The elliptic curves in class 133848z have rank $$2$$.

## Modular form 133848.2.a.g

sage: E.q_eigenform(10)

$$q - 2q^{5} - 4q^{7} - q^{11} - 6q^{17} + 8q^{19} + O(q^{20})$$

## 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. 