Properties

 Label 13680.a Number of curves $4$ Conductor $13680$ CM no Rank $2$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("13680.a1")

sage: E.isogeny_class()

Elliptic curves in class 13680.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
13680.a1 13680bh4 [0, 0, 0, -437763, 111482498] [2] 73728
13680.a2 13680bh3 [0, 0, 0, -31683, 1154882] [2] 73728
13680.a3 13680bh2 [0, 0, 0, -27363, 1741538] [2, 2] 36864
13680.a4 13680bh1 [0, 0, 0, -1443, 36002] [2] 18432 $$\Gamma_0(N)$$-optimal

Rank

sage: E.rank()

The elliptic curves in class 13680.a have rank $$2$$.

Modular form 13680.2.a.a

sage: E.q_eigenform(10)

$$q - q^{5} - 4q^{7} - 4q^{11} - 2q^{13} + 2q^{17} + q^{19} + O(q^{20})$$

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.