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SageMath
E = EllipticCurve("hm1")
E.isogeny_class()
Elliptic curves in class 216384hm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
216384.bd5 | 216384hm1 | \([0, -1, 0, 395071, -195888447]\) | \(221115865823/664731648\) | \(-20500975099721023488\) | \([2]\) | \(4718592\) | \(2.3889\) | \(\Gamma_0(N)\)-optimal |
216384.bd4 | 216384hm2 | \([0, -1, 0, -3619009, -2266350911]\) | \(169967019783457/26337394944\) | \(812271056392462270464\) | \([2, 2]\) | \(9437184\) | \(2.7355\) | |
216384.bd3 | 216384hm3 | \([0, -1, 0, -15912129, 22233837249]\) | \(14447092394873377/1439452851984\) | \(44394135835919152840704\) | \([2, 2]\) | \(18874368\) | \(3.0821\) | |
216384.bd2 | 216384hm4 | \([0, -1, 0, -55551169, -159340362047]\) | \(614716917569296417/19093020912\) | \(588847465699970383872\) | \([2]\) | \(18874368\) | \(3.0821\) | |
216384.bd1 | 216384hm5 | \([0, -1, 0, -248164289, 1504792275393]\) | \(54804145548726848737/637608031452\) | \(19664456198991733850112\) | \([2]\) | \(37748736\) | \(3.4286\) | |
216384.bd6 | 216384hm6 | \([0, -1, 0, 19650111, 107490751425]\) | \(27207619911317663/177609314617308\) | \(-5477645223306636530024448\) | \([2]\) | \(37748736\) | \(3.4286\) |
Rank
sage: E.rank()
The elliptic curves in class 216384hm have rank \(0\).
Complex multiplication
The elliptic curves in class 216384hm do not have complex multiplication.Modular form 216384.2.a.hm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.