Properties

 Label 111090.di Number of curves $8$ Conductor $111090$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("111090.di1")

sage: E.isogeny_class()

Elliptic curves in class 111090.di

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
111090.di1 111090dd8 [1, 0, 0, -3412590, -1526860650] [2] 7299072
111090.di2 111090dd5 [1, 0, 0, -3047580, -2048021928] [2] 2433024
111090.di3 111090dd6 [1, 0, 0, -1428840, 639791100] [2, 2] 3649536
111090.di4 111090dd3 [1, 0, 0, -1418260, 649983872] [2] 1824768
111090.di5 111090dd2 [1, 0, 0, -190980, -31833648] [2, 2] 1216512
111090.di6 111090dd4 [1, 0, 0, -42860, -79913400] [2] 2433024
111090.di7 111090dd1 [1, 0, 0, -21700, 431120] [2] 608256 $$\Gamma_0(N)$$-optimal
111090.di8 111090dd7 [1, 0, 0, 385630, 2154147762] [2] 7299072

Rank

sage: E.rank()

The elliptic curves in class 111090.di have rank $$1$$.

Modular form 111090.2.a.di

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + q^{10} + q^{12} + 2q^{13} - q^{14} + q^{15} + q^{16} + 6q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.