Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-698523568x-7082473116076\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-698523568xz^2-7082473116076z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-56580409035x-5162953160392326\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(99452, 30118230)$ | $6.3645869647238369331632777077$ | $\infty$ |
| $(-62467/4, 1066755/8)$ | $6.5315025200541455537970697206$ | $\infty$ |
| $(-15973, 0)$ | $0$ | $2$ |
Integral points
\( \left(-15973, 0\right) \), \((99452,\pm 30118230)\)
Invariants
| Conductor: | $N$ | = | \( 121296 \) | = | $2^{4} \cdot 3 \cdot 7 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $145079076275548680242331648$ | = | $2^{18} \cdot 3^{6} \cdot 7^{3} \cdot 19^{12} $ |
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| j-invariant: | $j$ | = | \( \frac{195607431345044517625}{752875610010048} \) | = | $2^{-6} \cdot 3^{-6} \cdot 5^{3} \cdot 7^{-3} \cdot 17^{3} \cdot 19^{-6} \cdot 31^{3} \cdot 2203^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8782937618090426998322165625$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7129270916658771604104707251$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0242979660857512$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.211097646709122$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $39.554151283615232931399699402$ |
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| Real period: | $\Omega$ | ≈ | $0.029373441132852746623300879188$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2\cdot( 2 \cdot 3 )\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $13.942098411470687395122862689 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.942098411 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.029373 \cdot 39.554151 \cdot 48}{2^2} \\ & \approx 13.942098411\end{aligned}$$
Modular invariants
Modular form 121296.2.a.cp
For more coefficients, see the Downloads section to the right.
| Modular degree: | 59719680 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}^{*}$ | additive | -1 | 4 | 18 | 6 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $19$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
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 |
| $3$ | 3Cs | 3.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 1608 \\ 186 & 2863 \end{array}\right),\left(\begin{array}{rr} 3590 & 4353 \\ 4389 & 3590 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 4126 & 3 \\ 1161 & 4480 \end{array}\right),\left(\begin{array}{rr} 4753 & 36 \\ 4752 & 37 \end{array}\right),\left(\begin{array}{rr} 3503 & 4752 \\ 2730 & 1391 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 1440 & 1819 \end{array}\right)$.
The torsion field $K:=\Q(E[4788])$ is a degree-$107226685440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4788\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 2527 = 7 \cdot 19^{2} \) |
| $3$ | split multiplicative | $4$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 17328 = 2^{4} \cdot 3 \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 121296cn
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 798e4, its twist by $76$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{19}) \) | \(\Z/6\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-57}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.90972.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{19})\) | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{19})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{-57})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.4.35325239798016.21 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.6488309350656.18 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.6488309350656.1 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.132414476544.2 | \(\Z/12\Z\) | not in database |
| $8$ | 8.0.132414476544.14 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.11341175160095532759059921456262866422267707654144.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.26882785564670892465919813822252720408338269995008.3 | \(\Z/18\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | split | ss | nonsplit | ord | ord | ss | add | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | 2,2 | 2 | 2 | 2 | 2,2 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 1 | 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.