# Properties

 Label 177450.hb Number of curves 8 Conductor 177450 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("177450.hb1")

sage: E.isogeny_class()

## Elliptic curves in class 177450.hb

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177450.hb1 177450dr8 [1, 1, 1, -169804672463, -26932316475430969] [2] 445906944
177450.hb2 177450dr6 [1, 1, 1, -10612798713, -420820204853469] [2, 2] 222953472
177450.hb3 177450dr7 [1, 1, 1, -10482372963, -431667192777969] [2] 445906944
177450.hb4 177450dr5 [1, 1, 1, -2097311213, -36909542328469] [2] 148635648
177450.hb5 177450dr3 [1, 1, 1, -671458213, -6405484770469] [4] 111476736
177450.hb6 177450dr2 [1, 1, 1, -174936213, -157577078469] [2, 2] 74317824
177450.hb7 177450dr1 [1, 1, 1, -108688213, 433752569531] [4] 37158912 $$\Gamma_0(N)$$-optimal
177450.hb8 177450dr4 [1, 1, 1, 687470787, -1249384340469] [2] 148635648

## Rank

sage: E.rank()

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

## Modular form 177450.2.a.hb

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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.