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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 16170.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16170.k1 | 16170m7 | \([1, 1, 0, -1247931437, 16967557264029]\) | \(1826870018430810435423307849/7641104625000000000\) | \(898968318026625000000000\) | \([2]\) | \(7962624\) | \(3.8052\) | |
| 16170.k2 | 16170m6 | \([1, 1, 0, -79206957, 256433414301]\) | \(467116778179943012100169/28800309694464000000\) | \(3388327635243995136000000\) | \([2, 2]\) | \(3981312\) | \(3.4586\) | |
| 16170.k3 | 16170m4 | \([1, 1, 0, -21450902, 3348842316]\) | \(9278380528613437145689/5328033205714065000\) | \(626837778619054033185000\) | \([2]\) | \(2654208\) | \(3.2559\) | |
| 16170.k4 | 16170m3 | \([1, 1, 0, -14981677, -17384644451]\) | \(3160944030998056790089/720291785342976000\) | \(84741608253815783424000\) | \([2]\) | \(1990656\) | \(3.1120\) | |
| 16170.k5 | 16170m2 | \([1, 1, 0, -14055822, -20207445516]\) | \(2610383204210122997209/12104550027662400\) | \(1424088206204453697600\) | \([2, 2]\) | \(1327104\) | \(2.9093\) | |
| 16170.k6 | 16170m1 | \([1, 1, 0, -14040142, -20254933964]\) | \(2601656892010848045529/56330588160\) | \(6627237366435840\) | \([2]\) | \(663552\) | \(2.5627\) | \(\Gamma_0(N)\)-optimal |
| 16170.k7 | 16170m5 | \([1, 1, 0, -6911622, -40724159076]\) | \(-310366976336070130009/5909282337130963560\) | \(-695221157681120731870440\) | \([2]\) | \(2654208\) | \(3.2559\) | |
| 16170.k8 | 16170m8 | \([1, 1, 0, 61913043, 1070949830301]\) | \(223090928422700449019831/4340371122724101696000\) | \(-510640322217367840432704000\) | \([2]\) | \(7962624\) | \(3.8052\) |
Rank
sage: E.rank()
The elliptic curves in class 16170.k have rank \(1\).
Complex multiplication
The elliptic curves in class 16170.k do not have complex multiplication.Modular form 16170.2.a.k
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
