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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 76230t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76230.c7 | 76230t1 | \([1, -1, 0, -541800, 153576000]\) | \(13619385906841/6048000\) | \(7810798276512000\) | \([2]\) | \(1105920\) | \(2.0081\) | \(\Gamma_0(N)\)-optimal |
| 76230.c6 | 76230t2 | \([1, -1, 0, -628920, 100938096]\) | \(21302308926361/8930250000\) | \(11533131830162250000\) | \([2, 2]\) | \(2211840\) | \(2.3546\) | |
| 76230.c5 | 76230t3 | \([1, -1, 0, -1603575, -593603235]\) | \(353108405631241/86318776320\) | \(111477934740555694080\) | \([2]\) | \(3317760\) | \(2.5574\) | |
| 76230.c8 | 76230t4 | \([1, -1, 0, 2093580, 739636596]\) | \(785793873833639/637994920500\) | \(-823950004210451464500\) | \([2]\) | \(4423680\) | \(2.7012\) | |
| 76230.c4 | 76230t5 | \([1, -1, 0, -4745340, -3907631700]\) | \(9150443179640281/184570312500\) | \(238366646622070312500\) | \([2]\) | \(4423680\) | \(2.7012\) | |
| 76230.c2 | 76230t6 | \([1, -1, 0, -23906295, -44980476579]\) | \(1169975873419524361/108425318400\) | \(140027825742226329600\) | \([2, 2]\) | \(6635520\) | \(2.9039\) | |
| 76230.c3 | 76230t7 | \([1, -1, 0, -22163895, -51816608739]\) | \(-932348627918877961/358766164249920\) | \(-463335009489764590812480\) | \([2]\) | \(13271040\) | \(3.2505\) | |
| 76230.c1 | 76230t8 | \([1, -1, 0, -382492215, -2879171871075]\) | \(4791901410190533590281/41160000\) | \(53156821604040000\) | \([2]\) | \(13271040\) | \(3.2505\) |
Rank
sage: E.rank()
The elliptic curves in class 76230t have rank \(1\).
Complex multiplication
The elliptic curves in class 76230t do not have complex multiplication.Modular form 76230.2.a.t
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
