Properties

 Label 36.a Number of curves $4$ Conductor $36$ CM $$\Q(\sqrt{-3})$$ Rank $0$ Graph Related objects

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

sage: E.isogeny_class()

Elliptic curves in class 36.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
36.a1 36a4 $$[0, 0, 0, -135, -594]$$ $$54000$$ $$5038848$$ $$$$ $$6$$ $$0.080464$$   $$-12$$
36.a2 36a2 $$[0, 0, 0, -15, 22]$$ $$54000$$ $$6912$$ $$$$ $$2$$ $$-0.46884$$   $$-12$$
36.a3 36a3 $$[0, 0, 0, 0, -27]$$ $$0$$ $$-314928$$ $$$$ $$3$$ $$-0.26611$$   $$-3$$
36.a4 36a1 $$[0, 0, 0, 0, 1]$$ $$0$$ $$-432$$ $$$$ $$1$$ $$-0.81542$$ $$\Gamma_0(N)$$-optimal $$-3$$

Rank

sage: E.rank()

The elliptic curves in class 36.a have rank $$0$$.

Complex multiplication

Each elliptic curve in class 36.a has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

Modular form36.2.a.a

sage: E.q_eigenform(10)

$$q - 4 q^{7} + 2 q^{13} + 8 q^{19} + O(q^{20})$$ 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.

$$\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with LMFDB labels. 