# Properties

 Label 134310ch Number of curves $6$ Conductor $134310$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("134310.r1")

sage: E.isogeny_class()

## Elliptic curves in class 134310ch

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
134310.r5 134310ch1 [1, 1, 0, -102247, -43363019] [2] 2211840 $$\Gamma_0(N)$$-optimal
134310.r4 134310ch2 [1, 1, 0, -2580327, -1593154251] [2, 2] 4423680
134310.r3 134310ch3 [1, 1, 0, -3548327, -290419851] [2, 2] 8847360
134310.r1 134310ch4 [1, 1, 0, -41261607, -102032965899] [2] 8847360
134310.r2 134310ch5 [1, 1, 0, -36678127, 85111578589] [2] 17694720
134310.r6 134310ch6 [1, 1, 0, 14093473, -2298056691] [2] 17694720

## Rank

sage: E.rank()

The elliptic curves in class 134310ch have rank $$0$$.

## Modular form 134310.2.a.r

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} - q^{12} + 2q^{13} - q^{15} + q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.