hillaxiom

Leaderboard ⫽ Torsion Rank Records

Three kinds of rank records for each rational torsion group: the best single curve, the best parametric family, and the best rank achieved by infinitely many curves.

Leaderboard

#1Noam Elkies
Individual Curve7/15
Parametric Family4/15
Infinite Family4/15
Total15/45
#2Andrej Dujella
Individual Curve5/15
Parametric Family6/15
Infinite Family4/15
Total15/45
#3Juan Carlos Peral
Individual Curve3/15
Parametric Family6/15
Infinite Family4/15
Total13/45
#4Zev Klagsbrun
Individual Curve7/15
Parametric Family0/15
Infinite Family0/15
Total7/45
#5Eroshkin
Individual Curve0/15
Parametric Family3/15
Infinite Family3/15
Total6/45
#21C. R. Hill
Individual Curve0/15
Parametric Family0/15
Infinite Family1/15
Total1/45

Torsion Rank Records

0

trivial

Individual Curve
≥32Elkies - Klagsbrun - Li (2026)
Parametric Family
≥18Elkies (2006)
Infinite Family
≥19Elkies (2006); Hill (2026)

ℤ/2ℤ

cyclic, order 2

Individual Curve
≥22Li (2026)
Parametric Family
≥11Elkies (2009); Dujella–Peral (2023)
Infinite Family
≥11Elkies (2007, 2009); Dujella–Peral (2023)

ℤ/3ℤ

cyclic, order 3

Individual Curve
≥15Elkies–Klagsbrun (2020)
Parametric Family
≥7Elkies (2007); Eroshkin (2023)
Infinite Family
≥8Eroshkin (2023)

ℤ/4ℤ

cyclic, order 4

Individual Curve
≥13Elkies–Klagsbrun (2020); Li (2026)
Parametric Family
≥6Dujella–Peral (2022)
Infinite Family
≥6Elkies (2007); Dujella–Peral (2021, 2022)

ℤ/5ℤ

cyclic, order 5

Individual Curve
≥9Klagsbrun (2020)
Parametric Family
≥4Eroshkin (2020)
Infinite Family
≥4Eroshkin (2009)

ℤ/6ℤ

cyclic, order 6

Individual Curve
≥9Klagsbrun (2020); Voznyy (2020)
Parametric Family
≥3Lecacheux (2001); Kihara (2006); Eroshkin (2008); Woo (2008); Dujella–Peral; MacLeod; Voznyy
Infinite Family
≥5Eroshkin (2009)

ℤ/7ℤ

cyclic, order 7

Individual Curve
≥6Klagsbrun (2020)
Parametric Family
≥1Kulesz (1998); Lecacheux (2003); Rabarison (2008); Harrache (2009); MacLeod (2014)
Infinite Family
≥3Dujella–Peral–Youmbai (2026)

ℤ/8ℤ

cyclic, order 8

Individual Curve
≥6Elkies; Dujella–MacLeod–Peral; Voznyy
Parametric Family
≥2Dujella–Peral (2012); MacLeod (2013); Dujella–Kazalicki–Peral (2021)
Infinite Family
≥3Dujella–Peral (2012); Dujella–Kazalicki–Peral (2021)

ℤ/9ℤ

cyclic, order 9

Individual Curve
≥4Fisher; van Beek; Dujella et al.; Voznyy (2026)
Parametric Family
≥0Kubert (1976)
Infinite Family
≥1Atkin–Morain; Kulesz; Rabarison; Gasull–Manosa–Xarles; Youmbai (2024)

ℤ/10ℤ

cyclic, order 10

Individual Curve
≥4Dujella; Elkies; Fisher
Parametric Family
≥0Kubert (1976)
Infinite Family
≥1Atkin–Morain; Kulesz; Rabarison; Voznyy (2021)

ℤ/12ℤ

cyclic, order 12

Individual Curve
≥4Fisher (2008)
Parametric Family
≥0Kubert (1976)
Infinite Family
≥1Suyama; Kulesz; Rabarison; Halbeisen et al. (2021)

ℤ/2ℤ × ℤ/2ℤ

Klein four

Individual Curve
≥15Elkies (2009)
Parametric Family
≥7Elkies (2007)
Infinite Family
≥8Elkies (2007)

ℤ/2ℤ × ℤ/4ℤ

order 8

Individual Curve
≥9Dujella–Peral; Klagsbrun
Parametric Family
≥4Dujella–Peral (2012); Jang (2026)
Infinite Family
≥6Jaswal (2026)

ℤ/2ℤ × ℤ/6ℤ

order 12

Individual Curve
≥6Elkies; Dujella–Peral–Tadić; Dujella–Peral
Parametric Family
≥2Dujella–Peral (2012, 2015, 2017); MacLeod (2013); Dujella–Kazalicki–Peral (2021)
Infinite Family
≥4Jaswal (2026)

ℤ/2ℤ × ℤ/8ℤ

order 16

Individual Curve
≥3Several rank-3 constructions
Parametric Family
≥0Kubert (1976)
Infinite Family
≥1Atkin–Morain; Kulesz; Lecacheux; Campbell–Goins; Rabarison

Other Records