Properties

 Label 444675.ca Number of curves $6$ Conductor $444675$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("444675.ca1")

sage: E.isogeny_class()

Elliptic curves in class 444675.ca

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
444675.ca1 444675ca6 [1, 1, 1, -1963984338, -33500010344844] [2] 283115520
444675.ca2 444675ca4 [1, 1, 1, -129699963, -460880182344] [2, 2] 141557760
444675.ca3 444675ca2 [1, 1, 1, -40023838, 90986690906] [2, 2] 70778880
444675.ca4 444675ca1 [1, 1, 1, -39282713, 94748641406] [2] 35389440 $$\Gamma_0(N)$$-optimal
444675.ca5 444675ca3 [1, 1, 1, 37794287, 402103554656] [2] 141557760
444675.ca6 444675ca5 [1, 1, 1, 269766412, -2739436385344] [2] 283115520

Rank

sage: E.rank()

The elliptic curves in class 444675.ca have rank $$1$$.

Modular form 444675.2.a.ca

sage: E.q_eigenform(10)

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

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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with LMFDB labels.