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Properties

Label 5077.a1
Conductor $5077$
Discriminant $5077$
j-invariant \( \frac{37933056}{5077} \)
CM no
Rank $3$
Torsion structure trivial

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This elliptic curve has smallest conductor amongst elliptic curves over $\Q$ of rank 3.

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+y=x^3-7x+6\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+yz^2=x^3-7xz^2+6z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-112x+400\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 0, 1, -7, 6])
 
gp: E = ellinit([0, 0, 1, -7, 6])
 
magma: E := EllipticCurve([0, 0, 1, -7, 6]);
 
oscar: E = elliptic_curve([0, 0, 1, -7, 6])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generators and heights

$P$ =  \(\left(1, 0\right)\) Copy content Toggle raw display \(\left(2, 0\right)\) Copy content Toggle raw display \(\left(0, 2\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.66820516565192793503314205089$$0.76704335533154620579545064655$$0.99090633315308797388259855289$

sage: E.gens()
 
magma: Generators(E);
 
gp: E.gen
 

Integral points

\( \left(-3, 0\right) \), \( \left(-3, -1\right) \), \( \left(-2, 3\right) \), \( \left(-2, -4\right) \), \( \left(-1, 3\right) \), \( \left(-1, -4\right) \), \( \left(0, 2\right) \), \( \left(0, -3\right) \), \( \left(1, 0\right) \), \( \left(1, -1\right) \), \( \left(2, 0\right) \), \( \left(2, -1\right) \), \( \left(3, 3\right) \), \( \left(3, -4\right) \), \( \left(4, 6\right) \), \( \left(4, -7\right) \), \( \left(8, 21\right) \), \( \left(8, -22\right) \), \( \left(11, 35\right) \), \( \left(11, -36\right) \), \( \left(14, 51\right) \), \( \left(14, -52\right) \), \( \left(21, 95\right) \), \( \left(21, -96\right) \), \( \left(37, 224\right) \), \( \left(37, -225\right) \), \( \left(52, 374\right) \), \( \left(52, -375\right) \), \( \left(93, 896\right) \), \( \left(93, -897\right) \), \( \left(342, 6324\right) \), \( \left(342, -6325\right) \), \( \left(406, 8180\right) \), \( \left(406, -8181\right) \), \( \left(816, 23309\right) \), \( \left(816, -23310\right) \) Copy content Toggle raw display

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: \( 5077 \)  =  $5077$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $5077 $  =  $5077 $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
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)
 
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $-0.56139014229398666466212500182\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $-0.56139014229398666466212500182\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $0.7202387335098172\dots$
Szpiro ratio: $2.0452836736223676\dots$

BSD invariants

Analytic rank: $3$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $0.41714355875838396981711954462\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
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);
 
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)
 
Torsion order: $1$
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Analytic order of Ш: $1$ ( rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L^{(3)}(E,1)/3! $ ≈ $ 1.7318499001193006897919750851 $
comment: Special L-value
 
r = E.rank();
 
E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 

BSD formula

$\displaystyle 1.731849900 \approx L^{(3)}(E,1)/3! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 4.151688 \cdot 0.417144 \cdot 1}{1^2} \approx 1.731849900$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
 
E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
 
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
 
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
 
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */
 
E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
 
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
 
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
 
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form   5077.2.a.a

\( q - 2 q^{2} - 3 q^{3} + 2 q^{4} - 4 q^{5} + 6 q^{6} - 4 q^{7} + 6 q^{9} + 8 q^{10} - 6 q^{11} - 6 q^{12} - 4 q^{13} + 8 q^{14} + 12 q^{15} - 4 q^{16} - 4 q^{17} - 12 q^{18} - 7 q^{19} + O(q^{20}) \) Copy content Toggle raw display

comment: q-expansion of modular form
 
sage: E.q_eigenform(20)
 
\\ actual modular form, use for small N
 
[mf,F] = mffromell(E)
 
Ser(mfcoefs(mf,20),q)
 
\\ or just the series
 
Ser(ellan(E,20),q)*q
 
magma: ModularForm(E);
 

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);
 
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data

This elliptic curve is semistable. There is only one prime $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$5077$ $1$ $I_{1}$ nonsplit multiplicative 1 1 1 1

comment: Local data
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$.

comment: mod p Galois image
 
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

gens = [[5079, 2, 5079, 3], [1, 2, 0, 1], [1, 0, 2, 1], [1, 1, 10153, 0], [10153, 2, 10152, 3]]
 
GL(2,Integers(10154)).subgroup(gens)
 
Gens := [[5079, 2, 5079, 3], [1, 2, 0, 1], [1, 0, 2, 1], [1, 1, 10153, 0], [10153, 2, 10152, 3]];
 
sub<GL(2,Integers(10154))|Gens>;
 

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

$\left(\begin{array}{rr} 5079 & 2 \\ 5079 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 10153 & 0 \end{array}\right),\left(\begin{array}{rr} 10153 & 2 \\ 10152 & 3 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[10154])$ is a degree-$1992802876951968$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10154\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$ Reduction type Serre weight Serre conductor
$5077$ nonsplit multiplicative $5078$ \( 1 \)

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 5077.a consists of this curve only.

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 $E(\Q)_{\rm tors}$ (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.

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.

$p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

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).