# Properties

 Label 862.d Number of curves 2 Conductor 862 CM no Rank 1 Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("862.d1")
sage: E.isogeny_class()

## Elliptic curves in class 862.d

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
862.d1 862e1 [1, 1, 1, -2460, 45949] 5 640 $$\Gamma_0(N)$$-optimal
862.d2 862e2 [1, 1, 1, 15380, -102531] 1 3200

## Rank

sage: E.rank()

The elliptic curves in class 862.d have rank $$1$$.

## Modular form862.2.a.d

sage: E.q_eigenform(10)
$$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} + q^{8} - 2q^{9} + q^{10} - 3q^{11} - q^{12} - 6q^{13} - 2q^{14} - q^{15} + q^{16} - 2q^{17} - 2q^{18} - 5q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.
{"ft_lang_label":"__label__en","ft_lang_prob":0.6235773,"math_prob":0.99891686,"size":874,"snap":"2019-13-2019-22","text_gpt3_token_len":325,"char_repetition_ratio":0.1632184,"word_repetition_ratio":0.0,"special_character_ratio":0.37528604,"punctuation_ratio":0.21857923,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99881446,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-05-24T05:38:14Z\",\"WARC-Record-ID\":\"<urn:uuid:17b99568-bde7-414a-b96c-b7df05c35db4>\",\"Content-Length\":\"36908\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:76c9be50-e0c8-4ff4-9dd4-2487bda0cd66>\",\"WARC-Concurrent-To\":\"<urn:uuid:e42b82de-2946-49cd-ad1a-47b4fe9bef6a>\",\"WARC-IP-Address\":\"35.241.19.59\",\"WARC-Target-URI\":\"http://www.lmfdb.org/EllipticCurve/Q/862/d/\",\"WARC-Payload-Digest\":\"sha1:ZBKGW2Z7J4VQYOUANVCNT6JIDLGXDQRJ\",\"WARC-Block-Digest\":\"sha1:PAWIE24PHGD4HG773HP35DLI7QDQABPX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-22/CC-MAIN-2019-22_segments_1558232257514.68_warc_CC-MAIN-20190524044320-20190524070320-00356.warc.gz\"}"}