Show commands:
SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 279174cc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
279174.cc4 | 279174cc1 | \([1, 1, 1, -173117, -45296701]\) | \(-23771111713777/22848457968\) | \(-551506230746199792\) | \([2]\) | \(4423680\) | \(2.1004\) | \(\Gamma_0(N)\)-optimal |
279174.cc3 | 279174cc2 | \([1, 1, 1, -3230737, -2235775669]\) | \(154502321244119857/55101928644\) | \(1330026604677626436\) | \([2, 2]\) | \(8847360\) | \(2.4470\) | |
279174.cc2 | 279174cc3 | \([1, 1, 1, -3696027, -1550310441]\) | \(231331938231569617/90942310746882\) | \(2195126300672305809858\) | \([2]\) | \(17694720\) | \(2.7936\) | |
279174.cc1 | 279174cc4 | \([1, 1, 1, -51687367, -143050742449]\) | \(632678989847546725777/80515134\) | \(1943439602469246\) | \([2]\) | \(17694720\) | \(2.7936\) |
Rank
sage: E.rank()
The elliptic curves in class 279174cc have rank \(1\).
Complex multiplication
The elliptic curves in class 279174cc do not have complex multiplication.Modular form 279174.2.a.cc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.