# Properties

 Label 41405.h Number of curves $3$ Conductor $41405$ CM no Rank $0$ Graph # Learn more

Show commands: SageMath
sage: E = EllipticCurve("h1")

sage: E.isogeny_class()

## Elliptic curves in class 41405.h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
41405.h1 41405d3 $$[0, -1, 1, -1087571, 457033527]$$ $$-250523582464/13671875$$ $$-7763837430248046875$$ $$[]$$ $$590976$$ $$2.3829$$
41405.h2 41405d1 $$[0, -1, 1, -11041, -491723]$$ $$-262144/35$$ $$-19875423821435$$ $$[]$$ $$65664$$ $$1.2843$$ $$\Gamma_0(N)$$-optimal
41405.h3 41405d2 $$[0, -1, 1, 71769, 1239006]$$ $$71991296/42875$$ $$-24347394181257875$$ $$[]$$ $$196992$$ $$1.8336$$

## Rank

sage: E.rank()

The elliptic curves in class 41405.h have rank $$0$$.

## Complex multiplication

The elliptic curves in class 41405.h do not have complex multiplication.

## Modular form 41405.2.a.h

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{4} - q^{5} - 2q^{9} + 3q^{11} + 2q^{12} + q^{15} + 4q^{16} - 3q^{17} + 2q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 