Properties

 Label 1290e Number of curves $4$ Conductor $1290$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

Elliptic curves in class 1290e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1290.c4 1290e1 $$[1, 0, 1, 666, 8632]$$ $$32740359775271/50724864000$$ $$-50724864000$$ $$[2]$$ $$1920$$ $$0.73942$$ $$\Gamma_0(N)$$-optimal
1290.c3 1290e2 $$[1, 0, 1, -4454, 86456]$$ $$9768641617435609/2396304000000$$ $$2396304000000$$ $$[2, 2]$$ $$3840$$ $$1.0860$$
1290.c2 1290e3 $$[1, 0, 1, -24454, -1401544]$$ $$1617141066657115609/89723013444000$$ $$89723013444000$$ $$[2]$$ $$7680$$ $$1.4326$$
1290.c1 1290e4 $$[1, 0, 1, -66374, 6575672]$$ $$32337636827233520089/3023437500000$$ $$3023437500000$$ $$[2]$$ $$7680$$ $$1.4326$$

Rank

sage: E.rank()

The elliptic curves in class 1290e have rank $$1$$.

Complex multiplication

The elliptic curves in class 1290e do not have complex multiplication.

Modular form1290.2.a.e

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} + q^{10} + 4q^{11} + q^{12} - 6q^{13} - q^{15} + q^{16} - 6q^{17} - q^{18} - 8q^{19} + O(q^{20})$$

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.