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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 2112.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2112.f1 | 2112h3 | \([0, -1, 0, -1254529, 541258849]\) | \(6663712298552914184/29403\) | \(963477504\) | \([4]\) | \(15360\) | \(1.8104\) | |
2112.f2 | 2112h2 | \([0, -1, 0, -78409, 8476489]\) | \(13015685560572352/864536409\) | \(3541141131264\) | \([2, 2]\) | \(7680\) | \(1.4638\) | |
2112.f3 | 2112h4 | \([0, -1, 0, -73569, 9563553]\) | \(-1343891598641864/421900912521\) | \(-13824849101488128\) | \([2]\) | \(15360\) | \(1.8104\) | |
2112.f4 | 2112h1 | \([0, -1, 0, -5204, 116478]\) | \(243578556889408/52089208083\) | \(3333709317312\) | \([2]\) | \(3840\) | \(1.1172\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2112.f have rank \(0\).
Complex multiplication
The elliptic curves in class 2112.f do not have complex multiplication.Modular form 2112.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.