Show commands:
SageMath
E = EllipticCurve("mc1")
E.isogeny_class()
Elliptic curves in class 418950.mc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
418950.mc1 | 418950mc3 | \([1, -1, 1, -5429592230, 153993632580897]\) | \(13209596798923694545921/92340\) | \(123744431455312500\) | \([2]\) | \(212336640\) | \(3.8145\) | \(\Gamma_0(N)\)-optimal* |
418950.mc2 | 418950mc4 | \([1, -1, 1, -343539230, 2343753864897]\) | \(3345930611358906241/165622259047500\) | \(221949667340019214804687500\) | \([2]\) | \(212336640\) | \(3.8145\) | |
418950.mc3 | 418950mc2 | \([1, -1, 1, -339349730, 2406210930897]\) | \(3225005357698077121/8526675600\) | \(11426560800583556250000\) | \([2, 2]\) | \(106168320\) | \(3.4679\) | \(\Gamma_0(N)\)-optimal* |
418950.mc4 | 418950mc1 | \([1, -1, 1, -20947730, 38573658897]\) | \(-758575480593601/40535043840\) | \(-54320835542527260000000\) | \([2]\) | \(53084160\) | \(3.1213\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 418950.mc have rank \(0\).
Complex multiplication
The elliptic curves in class 418950.mc do not have complex multiplication.Modular form 418950.2.a.mc
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.