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SageMath
E = EllipticCurve("hh1")
E.isogeny_class()
Elliptic curves in class 177450.hh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177450.hh1 | 177450dv7 | \([1, 1, 1, -27255563, 34445453531]\) | \(29689921233686449/10380965400750\) | \(782920894141073542968750\) | \([2]\) | \(31850496\) | \(3.2862\) | |
| 177450.hh2 | 177450dv4 | \([1, 1, 1, -24340313, 46210642031]\) | \(21145699168383889/2593080\) | \(195567216901875000\) | \([2]\) | \(10616832\) | \(2.7369\) | |
| 177450.hh3 | 177450dv6 | \([1, 1, 1, -11411813, -14448358969]\) | \(2179252305146449/66177562500\) | \(4991038348016601562500\) | \([2, 2]\) | \(15925248\) | \(2.9396\) | |
| 177450.hh4 | 177450dv3 | \([1, 1, 1, -11327313, -14678367969]\) | \(2131200347946769/2058000\) | \(155212076906250000\) | \([2]\) | \(7962624\) | \(2.5931\) | |
| 177450.hh5 | 177450dv2 | \([1, 1, 1, -1525313, 717532031]\) | \(5203798902289/57153600\) | \(4310461107225000000\) | \([2, 2]\) | \(5308416\) | \(2.3903\) | |
| 177450.hh6 | 177450dv5 | \([1, 1, 1, -342313, 1803526031]\) | \(-58818484369/18600435000\) | \(-1402824172842421875000\) | \([2]\) | \(10616832\) | \(2.7369\) | |
| 177450.hh7 | 177450dv1 | \([1, 1, 1, -173313, -9843969]\) | \(7633736209/3870720\) | \(291925408320000000\) | \([2]\) | \(2654208\) | \(2.0438\) | \(\Gamma_0(N)\)-optimal |
| 177450.hh8 | 177450dv8 | \([1, 1, 1, 3079937, -48619905469]\) | \(42841933504271/13565917968750\) | \(-1023126483512878417968750\) | \([2]\) | \(31850496\) | \(3.2862\) |
Rank
sage: E.rank()
The elliptic curves in class 177450.hh have rank \(1\).
Complex multiplication
The elliptic curves in class 177450.hh do not have complex multiplication.Modular form 177450.2.a.hh
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
