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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 13200cc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13200.cl4 | 13200cc1 | \([0, 1, 0, 5467, 573438]\) | \(72268906496/606436875\) | \(-151609218750000\) | \([2]\) | \(27648\) | \(1.4024\) | \(\Gamma_0(N)\)-optimal |
13200.cl3 | 13200cc2 | \([0, 1, 0, -78908, 7829688]\) | \(13584145739344/1195803675\) | \(4783214700000000\) | \([2]\) | \(55296\) | \(1.7490\) | |
13200.cl2 | 13200cc3 | \([0, 1, 0, -390533, 93880938]\) | \(-26348629355659264/24169921875\) | \(-6042480468750000\) | \([2]\) | \(82944\) | \(1.9517\) | |
13200.cl1 | 13200cc4 | \([0, 1, 0, -6249908, 6011849688]\) | \(6749703004355978704/5671875\) | \(22687500000000\) | \([2]\) | \(165888\) | \(2.2983\) |
Rank
sage: E.rank()
The elliptic curves in class 13200cc have rank \(1\).
Complex multiplication
The elliptic curves in class 13200cc do not have complex multiplication.Modular form 13200.2.a.cc
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.