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SageMath
E = EllipticCurve("ha1")
E.isogeny_class()
Elliptic curves in class 377520.ha
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
377520.ha1 | 377520ha6 | \([0, 1, 0, -17453080, -28070245612]\) | \(81025909800741361/11088090\) | \(80458661103575040\) | \([2]\) | \(15728640\) | \(2.6562\) | |
377520.ha2 | 377520ha3 | \([0, 1, 0, -1635960, 804210900]\) | \(66730743078481/60937500\) | \(442181625600000000\) | \([2]\) | \(7864320\) | \(2.3096\) | |
377520.ha3 | 377520ha4 | \([0, 1, 0, -1093880, -436284972]\) | \(19948814692561/231344100\) | \(1678705398333849600\) | \([2, 2]\) | \(7864320\) | \(2.3096\) | |
377520.ha4 | 377520ha5 | \([0, 1, 0, -222680, -1111290732]\) | \(-168288035761/73415764890\) | \(-532727832020145315840\) | \([2]\) | \(15728640\) | \(2.6562\) | |
377520.ha5 | 377520ha2 | \([0, 1, 0, -125880, 6284628]\) | \(30400540561/15210000\) | \(110368533749760000\) | \([2, 2]\) | \(3932160\) | \(1.9630\) | |
377520.ha6 | 377520ha1 | \([0, 1, 0, 29000, 770900]\) | \(371694959/249600\) | \(-1811175938457600\) | \([2]\) | \(1966080\) | \(1.6164\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 377520.ha have rank \(1\).
Complex multiplication
The elliptic curves in class 377520.ha do not have complex multiplication.Modular form 377520.2.a.ha
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.