This elliptic curve has smallest conductor amongst elliptic curves over $\Q$ of rank 3.
Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-7x+6\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-7xz^2+6z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-112x+400\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
| $P$ | = |
\(\left(1, 0\right)\)
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\(\left(2, 0\right)\)
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\(\left(0, 2\right)\)
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| $\hat{h}(P)$ | ≈ | $0.66820516565192793503314205089$ | $0.76704335533154620579545064655$ | $0.99090633315308797388259855289$ |
Integral points
\( \left(-3, 0\right) \), \( \left(-3, -1\right) \), \( \left(-2, 3\right) \), \( \left(-2, -4\right) \), \( \left(-1, 3\right) \), \( \left(-1, -4\right) \), \( \left(0, 2\right) \), \( \left(0, -3\right) \), \( \left(1, 0\right) \), \( \left(1, -1\right) \), \( \left(2, 0\right) \), \( \left(2, -1\right) \), \( \left(3, 3\right) \), \( \left(3, -4\right) \), \( \left(4, 6\right) \), \( \left(4, -7\right) \), \( \left(8, 21\right) \), \( \left(8, -22\right) \), \( \left(11, 35\right) \), \( \left(11, -36\right) \), \( \left(14, 51\right) \), \( \left(14, -52\right) \), \( \left(21, 95\right) \), \( \left(21, -96\right) \), \( \left(37, 224\right) \), \( \left(37, -225\right) \), \( \left(52, 374\right) \), \( \left(52, -375\right) \), \( \left(93, 896\right) \), \( \left(93, -897\right) \), \( \left(342, 6324\right) \), \( \left(342, -6325\right) \), \( \left(406, 8180\right) \), \( \left(406, -8181\right) \), \( \left(816, 23309\right) \), \( \left(816, -23310\right) \)
Invariants
| Conductor: | \( 5077 \) | = | $5077$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $5077 $ | = | $5077 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{37933056}{5077} \) | = | $2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 5077^{-1}$ | 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: | $-0.56139014229398666466212500182\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $-0.56139014229398666466212500182\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
| Analytic rank: | $3$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $0.41714355875838396981711954462\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $4.1516879830869330498841756835\dots$ | 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: | $1$ | 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: | $1$ (rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L^{(3)}(E,1)/3! $ ≈ $ 1.7318499001193006897919750851 $ | 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
$\displaystyle 1.731849900 \approx L^{(3)}(E,1)/3! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 4.151688 \cdot 0.417144 \cdot 1}{1^2} \approx 1.731849900$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1984 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There is only one prime of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $5077$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10154 = 2 \cdot 5077 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 5079 & 2 \\ 5079 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 10153 & 0 \end{array}\right),\left(\begin{array}{rr} 10153 & 2 \\ 10152 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[10154])$ is a degree-$51531188$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10154\Z)$.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 5077 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss | ss | ord | ord | nonsplit |
| $\lambda$-invariant(s) | 4,3 | 3,3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3,3 | 3,3 | 3 | 3 | ? |
| $\mu$-invariant(s) | 0,0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0,0 | 0 | 0 | ? |
An entry ? indicates that the invariants have not yet been computed.
Isogenies
This curve has no rational isogenies. Its isogeny class 5077.a consists of this curve only.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.20308.1 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.6.2093830264528.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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.
Additional information
Historical Information about the Gauss elliptic curve
In 1985, Buhler, Gross and Zagier used the celebrated Gross-Zagier Theorem on heights of Heegner points (see Gross, Benedict H.; Zagier, Don B. (1986), "Heegner points and derivatives of L-series", Inventiones Mathematicae 84 (2): 225–320, [10.1007/BF01388809]) to prove that the L-function of this curve has a zero of order 3 at its critical point $s=1$, thus establishing the first part of the Birch and Swinnerton-Dyer conjecture for this curve (see Math. Comp. 44 (1985), 473-481: [10.1090/S0025-5718-1985-0777279-X]). This was the first time that BSD had been established for any elliptic curve of rank $3$. To this day, it is not possible, even in principle, to establish BSD for any curve of rank $4$ or greater since there is no known method for rigourously establishing the value of the analytic rank when it is greater than $3$.
Via Goldfeld's method, which required the use of an L-function of analytic rank at least $3$, this elliptic curve also found an application in the context of obtaining explicit lower bounds for the class numbers of imaginary quadratic fields. This solved Gauss's Class Number Problem first posed by Gauss in 1801 is his book Disquisitiones Arithmeticae (Section V, Articles 303 and 304).