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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 2448p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2448.i2 | 2448p1 | \([0, 0, 0, -36795, 2715626]\) | \(1845026709625/793152\) | \(2368339181568\) | \([2]\) | \(4608\) | \(1.3346\) | \(\Gamma_0(N)\)-optimal |
2448.i3 | 2448p2 | \([0, 0, 0, -31035, 3594602]\) | \(-1107111813625/1228691592\) | \(-3668853434646528\) | \([2]\) | \(9216\) | \(1.6811\) | |
2448.i1 | 2448p3 | \([0, 0, 0, -108075, -10338982]\) | \(46753267515625/11591221248\) | \(34611201186988032\) | \([2]\) | \(13824\) | \(1.8839\) | |
2448.i4 | 2448p4 | \([0, 0, 0, 260565, -65561254]\) | \(655215969476375/1001033261568\) | \(-2989069302509862912\) | \([2]\) | \(27648\) | \(2.2305\) |
Rank
sage: E.rank()
The elliptic curves in class 2448p have rank \(0\).
Complex multiplication
The elliptic curves in class 2448p do not have complex multiplication.Modular form 2448.2.a.p
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.