# Properties

 Label 175.b Number of curves $3$ Conductor $175$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 175.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
175.b1 175b3 $$[0, -1, 1, -3283, -74657]$$ $$-250523582464/13671875$$ $$-213623046875$$ $$[]$$ $$144$$ $$0.93218$$
175.b2 175b1 $$[0, -1, 1, -33, 93]$$ $$-262144/35$$ $$-546875$$ $$[]$$ $$16$$ $$-0.16643$$ $$\Gamma_0(N)$$-optimal
175.b3 175b2 $$[0, -1, 1, 217, -282]$$ $$71991296/42875$$ $$-669921875$$ $$[]$$ $$48$$ $$0.38287$$

## Rank

sage: E.rank()

The elliptic curves in class 175.b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 175.b do not have complex multiplication.

## Modular form175.2.a.b

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{4} - q^{7} - 2q^{9} - 3q^{11} + 2q^{12} - 5q^{13} + 4q^{16} - 3q^{17} + 2q^{19} + O(q^{20})$$ ## 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. 