Properties

 Label 21675c Number of curves 8 Conductor 21675 CM no Rank 1 Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("21675.s1")

sage: E.isogeny_class()

Elliptic curves in class 21675c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
21675.s7 21675c1 [1, 1, 0, -150, 114375] [2] 30720 $$\Gamma_0(N)$$-optimal
21675.s6 21675c2 [1, 1, 0, -36275, 2607000] [2, 2] 61440
21675.s5 21675c3 [1, 1, 0, -72400, -3498125] [2, 2] 122880
21675.s4 21675c4 [1, 1, 0, -578150, 168962625] [2] 122880
21675.s8 21675c5 [1, 1, 0, 252725, -25931750] [2] 245760
21675.s2 21675c6 [1, 1, 0, -975525, -371070000] [2, 2] 245760
21675.s3 21675c7 [1, 1, 0, -794900, -512499375] [2] 491520
21675.s1 21675c8 [1, 1, 0, -15606150, -23736178125] [2] 491520

Rank

sage: E.rank()

The elliptic curves in class 21675c have rank $$1$$.

Modular form 21675.2.a.s

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - q^{6} - 3q^{8} + q^{9} + 4q^{11} + q^{12} + 2q^{13} - q^{16} + q^{18} + 4q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels.