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SageMath
E = EllipticCurve("hr1")
E.isogeny_class()
Elliptic curves in class 92400.hr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
92400.hr1 | 92400he4 | \([0, 1, 0, -5653208, 5171325588]\) | \(312196988566716625/25367712678\) | \(1623533611392000000\) | \([2]\) | \(1990656\) | \(2.5394\) | |
92400.hr2 | 92400he3 | \([0, 1, 0, -329208, 92229588]\) | \(-61653281712625/21875235228\) | \(-1400015054592000000\) | \([2]\) | \(995328\) | \(2.1928\) | |
92400.hr3 | 92400he2 | \([0, 1, 0, -145208, -10730412]\) | \(5290763640625/2291573592\) | \(146660709888000000\) | \([2]\) | \(663552\) | \(1.9901\) | |
92400.hr4 | 92400he1 | \([0, 1, 0, 30792, -1226412]\) | \(50447927375/39517632\) | \(-2529128448000000\) | \([2]\) | \(331776\) | \(1.6435\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 92400.hr have rank \(1\).
Complex multiplication
The elliptic curves in class 92400.hr do not have complex multiplication.Modular form 92400.2.a.hr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.