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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 76230ct
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
76230.cv2 | 76230ct1 | \([1, -1, 1, -12728, -548809]\) | \(4767078987/6860\) | \(328128528420\) | \([2]\) | \(138240\) | \(1.1122\) | \(\Gamma_0(N)\)-optimal |
76230.cv3 | 76230ct2 | \([1, -1, 1, -9098, -871153]\) | \(-1740992427/5882450\) | \(-281370213120150\) | \([2]\) | \(276480\) | \(1.4587\) | |
76230.cv1 | 76230ct3 | \([1, -1, 1, -50843, 3879307]\) | \(416832723/56000\) | \(1952699569128000\) | \([2]\) | \(414720\) | \(1.6615\) | |
76230.cv4 | 76230ct4 | \([1, -1, 1, 79837, 20449531]\) | \(1613964717/6125000\) | \(-213576515373375000\) | \([2]\) | \(829440\) | \(2.0080\) |
Rank
sage: E.rank()
The elliptic curves in class 76230ct have rank \(1\).
Complex multiplication
The elliptic curves in class 76230ct do not have complex multiplication.Modular form 76230.2.a.ct
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.