Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3+x^2-3x+1\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3+x^2z-3xz^2+z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-4320x+108432\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Torsion generators
\( \left(1, 0\right) \)
Integral points
\( \left(1, 0\right) \), \( \left(1, -1\right) \)
Invariants
Conductor: | \( 37 \) | = | $37$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $37 $ | = | $37 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{4096000}{37} \) | = | $2^{15} \cdot 5^{3} \cdot 37^{-1}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $-0.87653072914440633638846023140\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.87653072914440633638846023140\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.8826782160855301\dots$ | |||
Szpiro ratio: | $4.216523835223949\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $6.5311295574253750410258498692\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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Tamagawa product: | $ 1 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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Torsion order: | $3$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 0.72568106193615278233620554103 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 0.725681062 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 6.531130 \cdot 1.000000 \cdot 1}{3^2} \approx 0.725681062$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 6 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 3 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There is only one prime of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$37$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
---|---|---|
$3$ | 3B.1.1 | 27.72.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1998 = 2 \cdot 3^{3} \cdot 37 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 31 & 36 \\ 172 & 1231 \end{array}\right),\left(\begin{array}{rr} 257 & 1275 \\ 973 & 1312 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 562 & 9 \\ 1013 & 790 \end{array}\right),\left(\begin{array}{rr} 1945 & 54 \\ 1944 & 55 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right)$.
The torsion field $K:=\Q(E[1998])$ is a degree-$2656732608$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1998\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 37b
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.3.148.1 | \(\Z/6\Z\) | Not in database |
$3$ | 3.3.1369.1 | \(\Z/9\Z\) | 3.3.1369.1-37.1-a3 |
$6$ | 6.6.810448.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.0.50602347.2 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$6$ | 6.0.36963.1 | \(\Z/9\Z\) | Not in database |
$9$ | 9.9.6075640136512.1 | \(\Z/18\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$18$ | 18.0.129572244330949414435923.1 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$18$ | 18.0.726566512595229689293941092352.1 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.387675611964622937567232.1 | \(\Z/18\Z\) | Not in database |
$18$ | 18.18.1365795913530635497834467328.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
$p$ | 2 | 3 | 37 |
---|---|---|---|
Reduction type | ss | ord | split |
$\lambda$-invariant(s) | 0,5 | 0 | 1 |
$\mu$-invariant(s) | 0,0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Additional information
The twist $E^{(-139)}$ of this curve $E$ by $\Q(\sqrt{-139})$ was the earliest elliptic curve proved to have analytic rank $3$, and thus the one first used to give an effective lower bound for the Gauss class number problem. See Proposition 7.4 and Theorem 8.1 in Gross and Zagier's paper in Invent. Math. 85 (1986). [The modularity theorem was not yet proved in general, but was known for $E$ by explicit calculation of the modular curve [$X_0(37)$]; the vanishing of the Heegner trace for discriminant $-139$ was proved by Zagier (Notices of the AMS 31 (1984), 739-743).] Soon after Oesterlé obtained several improvements on the constant factor in the bound, one of which was to replace the curve $E^{(-139)}$ of conductor $139^2 37 = 714877$ by the [minimal rank-$3$ curve] of conductor $5077$, whose modularity had recently been proved by Mestre, Oesterlé, and Serre. See pages 34-36 of [Goldfeld's exposition] in the 1985 Bull. AMS.