Properties

 Label 6240b Number of curves $4$ Conductor $6240$ CM no Rank $1$ Graph

Related objects

Show commands: SageMath
E = EllipticCurve("b1")

E.isogeny_class()

Elliptic curves in class 6240b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6240.a3 6240b1 $$[0, -1, 0, -86, 240]$$ $$1111934656/342225$$ $$21902400$$ $$[2, 2]$$ $$1536$$ $$0.11317$$ $$\Gamma_0(N)$$-optimal
6240.a2 6240b2 $$[0, -1, 0, -536, -4440]$$ $$33324076232/1285245$$ $$658045440$$ $$[2]$$ $$3072$$ $$0.45975$$
6240.a1 6240b3 $$[0, -1, 0, -1256, 17556]$$ $$428320044872/73125$$ $$37440000$$ $$[2]$$ $$3072$$ $$0.45975$$
6240.a4 6240b4 $$[0, -1, 0, 239, 1345]$$ $$367061696/426465$$ $$-1746800640$$ $$[2]$$ $$3072$$ $$0.45975$$

Rank

sage: E.rank()

The elliptic curves in class 6240b have rank $$1$$.

Complex multiplication

The elliptic curves in class 6240b do not have complex multiplication.

Modular form6240.2.a.b

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} - 4 q^{7} + q^{9} - 4 q^{11} - q^{13} + q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})$$

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 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.