# Properties

 Label 3675.h Number of curves $2$ Conductor $3675$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 3675.h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3675.h1 3675a2 $$[0, -1, 1, -251533, -48572532]$$ $$-19539165184/46875$$ $$-4222266357421875$$ $$[]$$ $$24192$$ $$1.8776$$
3675.h2 3675a1 $$[0, -1, 1, 5717, -338157]$$ $$229376/675$$ $$-60800635546875$$ $$[]$$ $$8064$$ $$1.3283$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form3675.2.a.h

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. 