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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 450450o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450450.o4 | 450450o1 | \([1, -1, 0, -543065292, -4870353806384]\) | \(1555006827939811751684089/221961497899581440\) | \(2528280187012419840000000\) | \([2]\) | \(139345920\) | \(3.6982\) | \(\Gamma_0(N)\)-optimal* |
450450.o3 | 450450o2 | \([1, -1, 0, -592457292, -3931560062384]\) | \(2019051077229077416165369/582160888682835862400\) | \(6631176372652927245150000000\) | \([2]\) | \(278691840\) | \(4.0448\) | \(\Gamma_0(N)\)-optimal* |
450450.o2 | 450450o3 | \([1, -1, 0, -1256768667, 10259897608741]\) | \(19272683606216463573689449/7161126378530668544000\) | \(81569705155450896384000000000\) | \([2]\) | \(418037760\) | \(4.2475\) | \(\Gamma_0(N)\)-optimal* |
450450.o1 | 450450o4 | \([1, -1, 0, -17771840667, 911669042440741]\) | \(54497099771831721530744218729/16209843781074944000000\) | \(184640251818806784000000000000\) | \([2]\) | \(836075520\) | \(4.5941\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 450450o have rank \(0\).
Complex multiplication
The elliptic curves in class 450450o do not have complex multiplication.Modular form 450450.2.a.o
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.