Properties

 Label 54.a Number of curves $3$ Conductor $54$ CM no Rank $0$ Graph Related objects

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

sage: E.isogeny_class()

Elliptic curves in class 54.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54.a1 54a2 $$[1, -1, 0, -123, -667]$$ $$-1167051/512$$ $$-90699264$$ $$[]$$ $$18$$ $$0.23410$$
54.a2 54a3 $$[1, -1, 0, -3, 3]$$ $$-132651/2$$ $$-54$$ $$$$ $$18$$ $$-0.86451$$
54.a3 54a1 $$[1, -1, 0, 12, 8]$$ $$9261/8$$ $$-157464$$ $$$$ $$6$$ $$-0.31520$$ $$\Gamma_0(N)$$-optimal

Rank

sage: E.rank()

The elliptic curves in class 54.a have rank $$0$$.

Complex multiplication

The elliptic curves in class 54.a do not have complex multiplication.

Modular form54.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + 3 q^{5} - q^{7} - q^{8} - 3 q^{10} - 3 q^{11} - 4 q^{13} + q^{14} + q^{16} + 2 q^{19} + O(q^{20})$$ 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.

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with LMFDB labels. 