Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-145748x-19160494\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-145748xz^2-19160494z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-188888787x-893385330066\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-240, 1537\right) \) | $0.24205961194768469317473874672$ | $\infty$ |
| \( \left(-159, 79\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-240:1537:1]\) | $0.24205961194768469317473874672$ | $\infty$ |
| \([-159:79:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-8637, 306180\right) \) | $0.24205961194768469317473874672$ | $\infty$ |
| \( \left(-5721, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-255, 1327\right) \), \( \left(-255, -1073\right) \), \( \left(-240, 1537\right) \), \( \left(-240, -1298\right) \), \( \left(-159, 79\right) \), \( \left(570, 8827\right) \), \( \left(570, -9398\right) \), \( \left(705, 14767\right) \), \( \left(705, -15473\right) \), \( \left(930, 25027\right) \), \( \left(930, -25958\right) \), \( \left(30945, 5427727\right) \), \( \left(30945, -5458673\right) \)
\([-255:1327:1]\), \([-255:-1073:1]\), \([-240:1537:1]\), \([-240:-1298:1]\), \([-159:79:1]\), \([570:8827:1]\), \([570:-9398:1]\), \([705:14767:1]\), \([705:-15473:1]\), \([930:25027:1]\), \([930:-25958:1]\), \([30945:5427727:1]\), \([30945:-5458673:1]\)
\((-9177,\pm 259200)\), \((-8637,\pm 306180)\), \( \left(-5721, 0\right) \), \((20523,\pm 1968300)\), \((25383,\pm 3265920)\), \((33483,\pm 5506380)\), \((1114023,\pm 1175731200)\)
Invariants
| Conductor: | $N$ | = | \( 11130 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 53$ |
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| Minimal Discriminant: | $\Delta$ | = | $39748385577600000$ | = | $2^{10} \cdot 3^{14} \cdot 5^{5} \cdot 7^{2} \cdot 53 $ |
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| j-invariant: | $j$ | = | \( \frac{342394863219497382841}{39748385577600000} \) | = | $2^{-10} \cdot 3^{-14} \cdot 5^{-5} \cdot 7^{-2} \cdot 53^{-1} \cdot 2633^{3} \cdot 2657^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9169190572640921716142895842$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.9169190572640921716142895842$ |
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| $abc$ quality: | $Q$ | ≈ | $0.972740129433247$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.074645194571771$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.24205961194768469317473874672$ |
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| Real period: | $\Omega$ | ≈ | $0.24619254055547277094591668430$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 280 $ = $ 2\cdot( 2 \cdot 7 )\cdot5\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.1715289581890655580138333123 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.171528958 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.246193 \cdot 0.242060 \cdot 280}{2^2} \\ & \approx 4.171528958\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 134400 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $3$ | $14$ | $I_{14}$ | split multiplicative | -1 | 1 | 14 | 14 |
| $5$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $53$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6360 = 2^{3} \cdot 3 \cdot 5 \cdot 53 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 4562 & 1 \\ 4079 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3977 & 2386 \\ 2384 & 3975 \end{array}\right),\left(\begin{array}{rr} 1274 & 1 \\ 2543 & 0 \end{array}\right),\left(\begin{array}{rr} 4241 & 4 \\ 2122 & 9 \end{array}\right),\left(\begin{array}{rr} 3181 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 6357 & 4 \\ 6356 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[6360])$ is a degree-$22822791413760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6360\Z)$.
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 | $4$ | \( 265 = 5 \cdot 53 \) |
| $3$ | split multiplicative | $4$ | \( 3710 = 2 \cdot 5 \cdot 7 \cdot 53 \) |
| $5$ | split multiplicative | $6$ | \( 1113 = 3 \cdot 7 \cdot 53 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 530 = 2 \cdot 5 \cdot 53 \) |
| $53$ | nonsplit multiplicative | $54$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 11130p
consists of 2 curves linked by isogenies of
degree 2.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{265}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.152640.2 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1636170140160000.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | 53 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | split | split | nonsplit | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | 3 | 4 | 2 | 1 | 1 | 1 | 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 | 0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.