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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 177870cz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
177870.gk6 | 177870cz1 | \([1, 1, 1, -1698857245, 26950822819955]\) | \(2601656892010848045529/56330588160\) | \(11740555256120443146240\) | \([4]\) | \(79626240\) | \(3.7617\) | \(\Gamma_0(N)\)-optimal |
177870.gk5 | 177870cz2 | \([1, 1, 1, -1700754525, 26887606209267]\) | \(2610383204210122997209/12104550027662400\) | \(2522859126671768196973953600\) | \([2, 2]\) | \(159252480\) | \(4.1083\) | |
177870.gk4 | 177870cz3 | \([1, 1, 1, -1812782980, 23129897849477]\) | \(3160944030998056790089/720291785342976000\) | \(150124928259738143098404864000\) | \([4]\) | \(238878720\) | \(4.3110\) | |
177870.gk7 | 177870cz4 | \([1, 1, 1, -836306325, 54199674198627]\) | \(-310366976336070130009/5909282337130963560\) | \(-1231626689322723924873128556840\) | \([2]\) | \(318504960\) | \(4.4548\) | |
177870.gk3 | 177870cz5 | \([1, 1, 1, -2595559205, -4470286918525]\) | \(9278380528613437145689/5328033205714065000\) | \(1110481361928149982083251785000\) | \([2]\) | \(318504960\) | \(4.4548\) | |
177870.gk2 | 177870cz6 | \([1, 1, 1, -9584041860, -341360794643835]\) | \(467116778179943012100169/28800309694464000000\) | \(6002629093820487267127296000000\) | \([2, 2]\) | \(477757440\) | \(4.6576\) | |
177870.gk8 | 177870cz7 | \([1, 1, 1, 7491478140, -1425396766739835]\) | \(223090928422700449019831/4340371122724101696000\) | \(-904630479867722388764801530944000\) | \([2]\) | \(955514880\) | \(5.0041\) | |
177870.gk1 | 177870cz8 | \([1, 1, 1, -150999703940, -22584573716942203]\) | \(1826870018430810435423307849/7641104625000000000\) | \(1592577212451565811625000000000\) | \([2]\) | \(955514880\) | \(5.0041\) |
Rank
sage: E.rank()
The elliptic curves in class 177870cz have rank \(1\).
Complex multiplication
The elliptic curves in class 177870cz do not have complex multiplication.Modular form 177870.2.a.cz
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.