Properties

 Label 3675.i Number of curves $2$ Conductor $3675$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

Elliptic curves in class 3675.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3675.i1 3675i2 $$[0, 1, 1, -5133, 140144]$$ $$-19539165184/46875$$ $$-35888671875$$ $$[]$$ $$3456$$ $$0.90464$$
3675.i2 3675i1 $$[0, 1, 1, 117, 1019]$$ $$229376/675$$ $$-516796875$$ $$[]$$ $$1152$$ $$0.35533$$ $$\Gamma_0(N)$$-optimal

Rank

sage: E.rank()

The elliptic curves in class 3675.i have rank $$1$$.

Complex multiplication

The elliptic curves in class 3675.i do not have complex multiplication.

Modular form3675.2.a.i

sage: E.q_eigenform(10)

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

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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with LMFDB labels.