Properties

 Label 1568g Number of curves $4$ Conductor $1568$ CM $$\Q(\sqrt{-1})$$ Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

Elliptic curves in class 1568g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality CM discriminant
1568.e3 1568g1 [0, 0, 0, -49, 0] [2, 2] 192 $$\Gamma_0(N)$$-optimal -4
1568.e1 1568g2 [0, 0, 0, -539, -4802] [2] 384   -16
1568.e2 1568g3 [0, 0, 0, -539, 4802] [2] 384   -16
1568.e4 1568g4 [0, 0, 0, 196, 0] [2] 384   -4

Rank

sage: E.rank()

The elliptic curves in class 1568g have rank $$1$$.

Complex multiplication

Each elliptic curve in class 1568g has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-1})$$.

Modular form1568.2.a.g

sage: E.q_eigenform(10)

$$q + 2q^{5} - 3q^{9} - 6q^{13} - 2q^{17} + O(q^{20})$$

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.