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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1122k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1122.m4 | 1122k1 | \([1, 0, 0, -5847, -135063]\) | \(22106889268753393/4969545596928\) | \(4969545596928\) | \([4]\) | \(2688\) | \(1.1491\) | \(\Gamma_0(N)\)-optimal |
1122.m2 | 1122k2 | \([1, 0, 0, -87767, -10014615]\) | \(74768347616680342513/5615307472896\) | \(5615307472896\) | \([2, 2]\) | \(5376\) | \(1.4957\) | |
1122.m1 | 1122k3 | \([1, 0, 0, -1404247, -640608535]\) | \(306234591284035366263793/1727485056\) | \(1727485056\) | \([2]\) | \(10752\) | \(1.8423\) | |
1122.m3 | 1122k4 | \([1, 0, 0, -82007, -11384343]\) | \(-60992553706117024753/20624795251201152\) | \(-20624795251201152\) | \([2]\) | \(10752\) | \(1.8423\) |
Rank
sage: E.rank()
The elliptic curves in class 1122k have rank \(0\).
Complex multiplication
The elliptic curves in class 1122k do not have complex multiplication.Modular form 1122.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.