Properties

 Label 2475a Number of curves $2$ Conductor $2475$ CM no Rank $1$ Graph Related objects

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

sage: E.isogeny_class()

Elliptic curves in class 2475a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2475.j2 2475a1 $$[1, -1, 0, -42, -9]$$ $$19683/11$$ $$4640625$$ $$$$ $$320$$ $$-0.030545$$ $$\Gamma_0(N)$$-optimal
2475.j1 2475a2 $$[1, -1, 0, -417, 3366]$$ $$19034163/121$$ $$51046875$$ $$$$ $$640$$ $$0.31603$$

Rank

sage: E.rank()

The elliptic curves in class 2475a have rank $$1$$.

Complex multiplication

The elliptic curves in class 2475a do not have complex multiplication.

Modular form2475.2.a.a

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + 2 q^{7} - 3 q^{8} - q^{11} + 2 q^{13} + 2 q^{14} - q^{16} - 2 q^{17} - 6 q^{19} + O(q^{20})$$ 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels. 