# Properties

 Label 177450dv Number of curves $8$ Conductor $177450$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("177450.hh1")

sage: E.isogeny_class()

## Elliptic curves in class 177450dv

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177450.hh7 177450dv1 [1, 1, 1, -173313, -9843969] [2] 2654208 $$\Gamma_0(N)$$-optimal
177450.hh5 177450dv2 [1, 1, 1, -1525313, 717532031] [2, 2] 5308416
177450.hh4 177450dv3 [1, 1, 1, -11327313, -14678367969] [2] 7962624
177450.hh2 177450dv4 [1, 1, 1, -24340313, 46210642031] [2] 10616832
177450.hh6 177450dv5 [1, 1, 1, -342313, 1803526031] [2] 10616832
177450.hh3 177450dv6 [1, 1, 1, -11411813, -14448358969] [2, 2] 15925248
177450.hh1 177450dv7 [1, 1, 1, -27255563, 34445453531] [2] 31850496
177450.hh8 177450dv8 [1, 1, 1, 3079937, -48619905469] [2] 31850496

## Rank

sage: E.rank()

The elliptic curves in class 177450dv have rank $$1$$.

## Modular form 177450.2.a.hh

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} + q^{9} - q^{12} + q^{14} + q^{16} + 6q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.