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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 177450.cb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177450.cb1 | 177450iz8 | \([1, 1, 0, -1483957400, -22003535520000]\) | \(4791901410190533590281/41160000\) | \(3104241538125000000\) | \([2]\) | \(63700992\) | \(3.5895\) | |
| 177450.cb2 | 177450iz6 | \([1, 1, 0, -92749400, -343818168000]\) | \(1169975873419524361/108425318400\) | \(8177317229390400000000\) | \([2, 2]\) | \(31850496\) | \(3.2429\) | |
| 177450.cb3 | 177450iz7 | \([1, 1, 0, -85989400, -396052688000]\) | \(-932348627918877961/358766164249920\) | \(-27057746101515501645000000\) | \([2]\) | \(63700992\) | \(3.5895\) | |
| 177450.cb4 | 177450iz5 | \([1, 1, 0, -18410525, -29878254375]\) | \(9150443179640281/184570312500\) | \(13920088211059570312500\) | \([2]\) | \(21233664\) | \(3.0402\) | |
| 177450.cb5 | 177450iz3 | \([1, 1, 0, -6221400, -4541880000]\) | \(353108405631241/86318776320\) | \(6510066350161920000000\) | \([2]\) | \(15925248\) | \(2.8963\) | |
| 177450.cb6 | 177450iz2 | \([1, 1, 0, -2440025, 769135125]\) | \(21302308926361/8930250000\) | \(673509548003906250000\) | \([2, 2]\) | \(10616832\) | \(2.6936\) | |
| 177450.cb7 | 177450iz1 | \([1, 1, 0, -2102025, 1171693125]\) | \(13619385906841/6048000\) | \(456133450500000000\) | \([2]\) | \(5308416\) | \(2.3470\) | \(\Gamma_0(N)\)-optimal |
| 177450.cb8 | 177450iz4 | \([1, 1, 0, 8122475, 5659572625]\) | \(785793873833639/637994920500\) | \(-48116869128495070312500\) | \([2]\) | \(21233664\) | \(3.0402\) |
Rank
sage: E.rank()
The elliptic curves in class 177450.cb have rank \(0\).
Complex multiplication
The elliptic curves in class 177450.cb do not have complex multiplication.Modular form 177450.2.a.cb
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
