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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 12705.n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12705.n1 | 12705m5 | \([1, 0, 1, -368953203, 2727722241781]\) | \(3135316978843283198764801/571725\) | \(1012845712725\) | \([2]\) | \(921600\) | \(3.0981\) | |
12705.n2 | 12705m4 | \([1, 0, 1, -23059578, 42619209631]\) | \(765458482133960722801/326869475625\) | \(579069215107700625\) | \([2, 2]\) | \(460800\) | \(2.7515\) | |
12705.n3 | 12705m6 | \([1, 0, 1, -22945233, 43062822493]\) | \(-754127868744065783521/15825714261328125\) | \(-28036218182512714453125\) | \([2]\) | \(921600\) | \(3.0981\) | |
12705.n4 | 12705m3 | \([1, 0, 1, -3078848, -1096801477]\) | \(1821931919215868881/761147600816295\) | \(1348419404849716386495\) | \([2]\) | \(460800\) | \(2.7515\) | |
12705.n5 | 12705m2 | \([1, 0, 1, -1448373, 658894003]\) | \(189674274234120481/3859869269025\) | \(6837993862103198025\) | \([2, 2]\) | \(230400\) | \(2.4049\) | |
12705.n6 | 12705m1 | \([1, 0, 1, 4232, 30787601]\) | \(4733169839/231139696095\) | \(-409478071153754295\) | \([2]\) | \(115200\) | \(2.0584\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12705.n have rank \(0\).
Complex multiplication
The elliptic curves in class 12705.n do not have complex multiplication.Modular form 12705.2.a.n
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.