Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-163980x-22341014\)
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(homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-163980xz^2-22341014z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-212517459x-1041704785170\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = |
\(\left(-178, 1191\right)\)
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$\hat{h}(P)$ | ≈ | $2.0940889912565210678809308277$ |
Torsion generators
\( \left(-\frac{649}{4}, \frac{645}{8}\right) \)
Integral points
\( \left(-178, 1191\right) \), \( \left(-178, -1014\right) \), \( \left(5240, 375537\right) \), \( \left(5240, -380778\right) \)
Invariants
Conductor: | \( 6762 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $66838136399760096 $ | = | $2^{5} \cdot 3^{8} \cdot 7^{12} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{4144806984356137}{568114785504} \) | = | $2^{-5} \cdot 3^{-8} \cdot 7^{-6} \cdot 11^{3} \cdot 17^{3} \cdot 23^{-1} \cdot 859^{3}$ | 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: | $1.9552910910376623324075446393\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.98233601651000567985486826758\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9797147588676234\dots$ | |||
Szpiro ratio: | $5.401484736518478\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.0940889912565210678809308277\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.23940413376664325655356661758\dots$ | 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: | $ 32 $ = $ 1\cdot2^{3}\cdot2^{2}\cdot1 $ | 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: | $2$ | 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'(E,1) $ ≈ $ 4.0106684878562496849771279564 $ | 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
$\displaystyle 4.010668488 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.239404 \cdot 2.094089 \cdot 32}{2^2} \approx 4.010668488$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 92160 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
$3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
$7$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
$23$ | $1$ | $I_{1}$ | nonsplit 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 |
---|---|---|
$2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1288 = 2^{3} \cdot 7 \cdot 23 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 562 & 1 \\ 1007 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 643 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1285 & 4 \\ 1284 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 185 & 4 \\ 370 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 484 & 809 \\ 161 & 1128 \end{array}\right)$.
The torsion field $K:=\Q(E[1288])$ is a degree-$68942168064$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1288\Z)$.
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 |
---|---|---|---|
$2$ | nonsplit multiplicative | $4$ | \( 1127 = 7^{2} \cdot 23 \) |
$3$ | split multiplicative | $4$ | \( 2254 = 2 \cdot 7^{2} \cdot 23 \) |
$5$ | good | $2$ | \( 3381 = 3 \cdot 7^{2} \cdot 23 \) |
$7$ | additive | $32$ | \( 138 = 2 \cdot 3 \cdot 23 \) |
$23$ | nonsplit multiplicative | $24$ | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 6762n
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 966a2, its twist by $-7$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{46}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$4$ | 4.0.36064.1 | \(\Z/4\Z\) | not in database |
$8$ | deg 8 | \(\Z/6\Z\) | not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.44033523122176.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$16$ | deg 16 | \(\Z/8\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | split | ord | add | ss | ord | ss | ord | nonsplit | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 5 | 10 | 1 | - | 1,3 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 0 | 0 | 0 | - | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.