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SageMath
E = EllipticCurve("fc1")
E.isogeny_class()
Elliptic curves in class 222768.fc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 222768.fc1 | 222768bu6 | \([0, 0, 0, -5843379, -5432921678]\) | \(7389727131216686257/6115533215337\) | \(18260884332464836608\) | \([2]\) | \(6291456\) | \(2.6243\) | |
| 222768.fc2 | 222768bu4 | \([0, 0, 0, -445539, -44797790]\) | \(3275619238041697/1605271262049\) | \(4793314304138121216\) | \([2, 2]\) | \(3145728\) | \(2.2778\) | |
| 222768.fc3 | 222768bu2 | \([0, 0, 0, -237459, 44052370]\) | \(495909170514577/6224736609\) | \(18586963918688256\) | \([2, 2]\) | \(1572864\) | \(1.9312\) | |
| 222768.fc4 | 222768bu1 | \([0, 0, 0, -236739, 44335618]\) | \(491411892194497/78897\) | \(235585179648\) | \([2]\) | \(786432\) | \(1.5846\) | \(\Gamma_0(N)\)-optimal |
| 222768.fc5 | 222768bu3 | \([0, 0, 0, -40899, 114774658]\) | \(-2533811507137/1904381781393\) | \(-5686453529130995712\) | \([2]\) | \(3145728\) | \(2.2778\) | |
| 222768.fc6 | 222768bu5 | \([0, 0, 0, 1623021, -343084142]\) | \(158346567380527343/108665074944153\) | \(-324472175142041751552\) | \([2]\) | \(6291456\) | \(2.6243\) |
Rank
sage: E.rank()
The elliptic curves in class 222768.fc have rank \(1\).
Complex multiplication
The elliptic curves in class 222768.fc do not have complex multiplication.Modular form 222768.2.a.fc
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
