# Properties

 Label 21175r Number of curves $3$ Conductor $21175$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("r1")

sage: E.isogeny_class()

## Elliptic curves in class 21175r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21175.u2 21175r1 $$[0, -1, 1, -4033, -108032]$$ $$-262144/35$$ $$-968822421875$$ $$[]$$ $$21600$$ $$1.0325$$ $$\Gamma_0(N)$$-optimal
21175.u3 21175r2 $$[0, -1, 1, 26217, 270093]$$ $$71991296/42875$$ $$-1186807466796875$$ $$[]$$ $$64800$$ $$1.5818$$
21175.u1 21175r3 $$[0, -1, 1, -397283, 100957218]$$ $$-250523582464/13671875$$ $$-378446258544921875$$ $$[]$$ $$194400$$ $$2.1311$$

## Rank

sage: E.rank()

The elliptic curves in class 21175r have rank $$1$$.

## Complex multiplication

The elliptic curves in class 21175r do not have complex multiplication.

## Modular form 21175.2.a.r

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 