Show commands:
SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 363090.be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
363090.be1 | 363090be7 | \([1, 1, 0, -652331356107, 202791574383463101]\) | \(260939746299651996897062684320310569/4502310731746516416000000\) | \(529692355279245909825984000000\) | \([2]\) | \(3057647616\) | \(5.2447\) | |
363090.be2 | 363090be6 | \([1, 1, 0, -40811356107, 3161971119463101]\) | \(63896717795469435410864800310569/264596810010624000000000000\) | \(31129550100939902976000000000000\) | \([2, 2]\) | \(1528823808\) | \(4.8982\) | |
363090.be3 | 363090be8 | \([1, 1, 0, -21191536587, 6207779369495229]\) | \(-8945874824846901999181300606249/136327175537109375000000000000\) | \(-16038755874765380859375000000000000\) | \([2]\) | \(3057647616\) | \(5.2447\) | |
363090.be4 | 363090be4 | \([1, 1, 0, -8554086972, 241636399716384]\) | \(588378042102789360957899335609/126092479227795814426383600\) | \(14834654088670949771449604156400\) | \([2]\) | \(1019215872\) | \(4.6954\) | |
363090.be5 | 363090be3 | \([1, 1, 0, -3817594827, -4820624893251]\) | \(52300395461270777993673352489/30181158775846600704000000\) | \(3550783148819576726224896000000\) | \([2]\) | \(764411904\) | \(4.5516\) | |
363090.be6 | 363090be2 | \([1, 1, 0, -2740824972, -51932168631216]\) | \(19354385182631020519933543609/1297852417956067777440000\) | \(152691039120113417948038560000\) | \([2, 2]\) | \(509607936\) | \(4.3489\) | |
363090.be7 | 363090be1 | \([1, 1, 0, -2695102092, -53854020157104]\) | \(18401835147394456911544300729/91846771642205798400\) | \(10805680836933869975961600\) | \([2]\) | \(254803968\) | \(4.0023\) | \(\Gamma_0(N)\)-optimal |
363090.be8 | 363090be5 | \([1, 1, 0, 2340870948, -222501324864384]\) | \(12057794750690459080173411911/188765599666333542693750000\) | \(-22208084035144474964376993750000\) | \([2]\) | \(1019215872\) | \(4.6954\) |
Rank
sage: E.rank()
The elliptic curves in class 363090.be have rank \(1\).
Complex multiplication
The elliptic curves in class 363090.be do not have complex multiplication.Modular form 363090.2.a.be
sage: E.q_eigenform(10)
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.