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SageMath
E = EllipticCurve("fh1")
E.isogeny_class()
Elliptic curves in class 270480fh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
270480.fh4 | 270480fh1 | \([0, 1, 0, -595856, 176807700]\) | \(48551226272641/9273600\) | \(4468857923174400\) | \([2]\) | \(2359296\) | \(2.0038\) | \(\Gamma_0(N)\)-optimal |
270480.fh3 | 270480fh2 | \([0, 1, 0, -658576, 137243924]\) | \(65553197996161/20996010000\) | \(10117773641687040000\) | \([2, 2]\) | \(4718592\) | \(2.3504\) | |
270480.fh6 | 270480fh3 | \([0, 1, 0, 1865904, 938008980]\) | \(1490881681033919/1650501562500\) | \(-795360699705600000000\) | \([2]\) | \(9437184\) | \(2.6970\) | |
270480.fh2 | 270480fh4 | \([0, 1, 0, -4186576, -3194599276]\) | \(16840406336564161/604708416900\) | \(291403122851299737600\) | \([2, 2]\) | \(9437184\) | \(2.6970\) | |
270480.fh5 | 270480fh5 | \([0, 1, 0, 1575824, -11305753516]\) | \(898045580910239/115117148363070\) | \(-55473837620292904673280\) | \([2]\) | \(18874368\) | \(3.0435\) | |
270480.fh1 | 270480fh6 | \([0, 1, 0, -66396976, -208264961836]\) | \(67176973097223766561/91487391870\) | \(44086887080401428480\) | \([2]\) | \(18874368\) | \(3.0435\) |
Rank
sage: E.rank()
The elliptic curves in class 270480fh have rank \(1\).
Complex multiplication
The elliptic curves in class 270480fh do not have complex multiplication.Modular form 270480.2.a.fh
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.