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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 167310w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
167310.fr7 | 167310w1 | \([1, -1, 1, -17369852, 26658524351]\) | \(164711681450297281/8097103872000\) | \(28491633731768942592000\) | \([4]\) | \(18579456\) | \(3.0680\) | \(\Gamma_0(N)\)-optimal |
167310.fr6 | 167310w2 | \([1, -1, 1, -48519932, -95649149761]\) | \(3590017885052913601/954068544000000\) | \(3357122736766353984000000\) | \([2, 2]\) | \(37158912\) | \(3.4146\) | |
167310.fr3 | 167310w3 | \([1, -1, 1, -1389920252, 19945296829631]\) | \(84392862605474684114881/11228954880\) | \(39511814926550503680\) | \([4]\) | \(55738368\) | \(3.6173\) | |
167310.fr8 | 167310w4 | \([1, -1, 1, 122318788, -618825645889]\) | \(57519563401957999679/80296734375000000\) | \(-282543633110705484375000000\) | \([2]\) | \(74317824\) | \(3.7611\) | |
167310.fr5 | 167310w5 | \([1, -1, 1, -717759932, -7400537597761]\) | \(11621808143080380273601/1335706803288000\) | \(4700009980594904286168000\) | \([2]\) | \(74317824\) | \(3.7611\) | |
167310.fr2 | 167310w6 | \([1, -1, 1, -1390041932, 19941630075839]\) | \(84415028961834287121601/30783551683856400\) | \(108319430428990751919920400\) | \([2, 2]\) | \(111476736\) | \(3.9639\) | |
167310.fr4 | 167310w7 | \([1, -1, 1, -1189543712, 25895946014111]\) | \(-52902632853833942200321/51713453577420277500\) | \(-181966392135310731928013677500\) | \([2]\) | \(222953472\) | \(4.3104\) | |
167310.fr1 | 167310w8 | \([1, -1, 1, -1592487032, 13752640434719]\) | \(126929854754212758768001/50235797102795981820\) | \(176766897634325236684994425020\) | \([2]\) | \(222953472\) | \(4.3104\) |
Rank
sage: E.rank()
The elliptic curves in class 167310w have rank \(0\).
Complex multiplication
The elliptic curves in class 167310w do not have complex multiplication.Modular form 167310.2.a.w
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 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.