# Properties

 Label 5415.h Number of curves $2$ Conductor $5415$ CM no Rank $2$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 5415.h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5415.h1 5415c2 $$[1, 1, 0, -27443, 75438]$$ $$48587168449/28048275$$ $$1319555807905275$$ $$$$ $$28800$$ $$1.5909$$
5415.h2 5415c1 $$[1, 1, 0, 6852, 13707]$$ $$756058031/438615$$ $$-20635029094815$$ $$$$ $$14400$$ $$1.2443$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 5415.h have rank $$2$$.

## Complex multiplication

The elliptic curves in class 5415.h do not have complex multiplication.

## Modular form5415.2.a.h

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 2q^{7} - 3q^{8} + q^{9} - q^{10} - 6q^{11} + q^{12} - 2q^{14} + q^{15} - q^{16} - 6q^{17} + q^{18} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 