Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2+539995x+756685122\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z+539995xz^2+756685122z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+8639925x+48436487750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(314, 30780\right)\) | \(\left(-82, 26721\right)\) |
$\hat{h}(P)$ | ≈ | $0.94187630529329829667055416559$ | $1.0698666555935399406452278283$ |
Torsion generators
\( \left(-\frac{2869}{4}, \frac{2865}{8}\right) \)
Integral points
\( \left(-621, 13785\right) \), \( \left(-621, -13165\right) \), \( \left(-445, 20913\right) \), \( \left(-445, -20469\right) \), \( \left(-82, 26721\right) \), \( \left(-82, -26640\right) \), \( \left(275, 30291\right) \), \( \left(275, -30567\right) \), \( \left(314, 30780\right) \), \( \left(314, -31095\right) \), \( \left(800, 40833\right) \), \( \left(800, -41634\right) \), \( \left(1304, 59985\right) \), \( \left(1304, -61290\right) \), \( \left(2564, 136530\right) \), \( \left(2564, -139095\right) \), \( \left(6254, 495585\right) \), \( \left(6254, -501840\right) \), \( \left(6939, 578405\right) \), \( \left(6939, -585345\right) \), \( \left(17558, 2319921\right) \), \( \left(17558, -2337480\right) \), \( \left(20939, 3021405\right) \), \( \left(20939, -3042345\right) \), \( \left(2475314, 3893205780\right) \), \( \left(2475314, -3895681095\right) \)
Invariants
Conductor: | \( 121275 \) | = | $3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-257516966101376953125 $ | = | $-1 \cdot 3^{7} \cdot 5^{10} \cdot 7^{7} \cdot 11^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{12994449551}{192163125} \) | = | $3^{-1} \cdot 5^{-4} \cdot 7^{-1} \cdot 11^{-4} \cdot 2351^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.5971256162007012529196326191\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $0.27014544112193956736895396230\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9226712134651223\dots$ | |||
Szpiro ratio: | $4.648323362285934\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.92295716382867884440323687317\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.12975880901176629545076826945\dots$ | 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: | $ 256 $ = $ 2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{2} $ | 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: | $2$ | 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);
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Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 7.6647566302263701485613888427 $ | 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
$\displaystyle 7.664756630 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.129759 \cdot 0.922957 \cdot 256}{2^2} \approx 7.664756630$
Modular invariants
Modular form 121275.2.a.br
For more coefficients, see the Downloads section to the right.
Modular degree: | 3538944 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$5$ | $4$ | $I_{4}^{*}$ | Additive | 1 | 2 | 10 | 4 |
$7$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$11$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3076 & 9239 \\ 6137 & 9234 \end{array}\right),\left(\begin{array}{rr} 9233 & 8 \\ 9232 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5779 & 5778 \\ 8098 & 3475 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3952 & 9237 \\ 1315 & 9238 \end{array}\right),\left(\begin{array}{rr} 2521 & 8 \\ 844 & 33 \end{array}\right),\left(\begin{array}{rr} 5543 & 9232 \\ 3692 & 9207 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 9234 & 9235 \end{array}\right),\left(\begin{array}{rr} 5776 & 1163 \\ 5779 & 5808 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$19619905536000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 121275.br
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1155.d4, its twist by $105$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-21}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{15}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-35}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{15}, \sqrt{-21})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.3512980316160000.56 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | ord | add | add | add | split | ord | ord | ord | ss | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | 7 | - | - | - | 3 | 2 | 4 | 2 | 2,2 | 2 | 2,2 | 2 | 2 | 2 | 4 |
$\mu$-invariant(s) | 1 | - | - | - | 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.