# Properties

 Label 231231.cb Number of curves $6$ Conductor $231231$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("231231.cb1")

sage: E.isogeny_class()

## Elliptic curves in class 231231.cb

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
231231.cb1 231231cb3 [1, 1, 0, -40696779, 99911269242] [2] 11796480
231231.cb2 231231cb6 [1, 1, 0, -17840484, -28093000767] [2] 23592960
231231.cb3 231231cb4 [1, 1, 0, -2810469, 1212522480] [2, 2] 11796480
231231.cb4 231231cb2 [1, 1, 0, -2543664, 1560169395] [2, 2] 5898240
231231.cb5 231231cb1 [1, 1, 0, -142419, 29615832] [2] 2949120 $$\Gamma_0(N)$$-optimal
231231.cb6 231231cb5 [1, 1, 0, 7950666, 8273979267] [2] 23592960

## Rank

sage: E.rank()

The elliptic curves in class 231231.cb have rank $$0$$.

## Modular form 231231.2.a.cb

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.