Properties

 Label 258570ef Number of curves 8 Conductor 258570 CM no Rank 1 Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("258570.ef1")

sage: E.isogeny_class()

Elliptic curves in class 258570ef

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
258570.ef7 258570ef1 [1, -1, 1, -3188238098, -66720369131503] [2] 371589120 $$\Gamma_0(N)$$-optimal
258570.ef6 258570ef2 [1, -1, 1, -8452601618, 210391515070481] [2, 2] 743178240
258570.ef5 258570ef3 [1, -1, 1, -39230827538, 2971343928843281] [2] 1114767360
258570.ef4 258570ef4 [1, -1, 1, -123768737618, 16757519007800081] [2] 1486356480
258570.ef8 258570ef5 [1, -1, 1, 22633718062, 1398361438905617] [2] 1486356480
258570.ef2 258570ef6 [1, -1, 1, -626556946658, 190892678863373777] [2, 2] 2229534720
258570.ef1 258570ef7 [1, -1, 1, -10024911009158, 12217124280120623777L] [2] 4459069440
258570.ef3 258570ef8 [1, -1, 1, -625420790078, 191619465502646081] [2] 4459069440

Rank

sage: E.rank()

The elliptic curves in class 258570ef have rank $$1$$.

Modular form 258570.2.a.ef

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{5} + 4q^{7} + q^{8} - q^{10} + 4q^{14} + q^{16} - q^{17} + 4q^{19} + O(q^{20})$$

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.