Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-43203x+1455298\)
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(homogenize, simplify) |
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\(y^2z=x^3-43203xz^2+1455298z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-43203x+1455298\)
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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 |
|---|---|---|
| $(33, 256)$ | $2.1788407163348828267805222208$ | $\infty$ |
| $(-177, 1886)$ | $3.4589888566140248427153065841$ | $\infty$ |
| $(-223, 0)$ | $0$ | $2$ |
Integral points
\( \left(-223, 0\right) \), \((-177,\pm 1886)\), \((2,\pm 1170)\), \((33,\pm 256)\), \((191,\pm 414)\), \((2081,\pm 94464)\), \((6018,\pm 466574)\)
Invariants
| Conductor: | $N$ | = | \( 115920 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $4245925920768000$ | = | $2^{22} \cdot 3^{7} \cdot 5^{3} \cdot 7 \cdot 23^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{2986606123201}{1421952000} \) | = | $2^{-10} \cdot 3^{-1} \cdot 5^{-3} \cdot 7^{-1} \cdot 23^{-2} \cdot 14401^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6926920074508844396080913242$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.45023868255688428449323658428$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9103137555231284$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7420364912936224$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.0767292276208797904716736804$ |
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| Real period: | $\Omega$ | ≈ | $0.39031913370085075374653350515$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $11.048731286241889863009371228 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.048731286 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.390319 \cdot 7.076729 \cdot 16}{2^2} \\ & \approx 11.048731286\end{aligned}$$
Modular invariants
Modular form 115920.2.a.bh
For more coefficients, see the Downloads section to the right.
| Modular degree: | 737280 |
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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 not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{14}^{*}$ | additive | -1 | 4 | 22 | 10 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $23$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 \( 19320 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9661 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 5522 & 1 \\ 16559 & 0 \end{array}\right),\left(\begin{array}{rr} 7249 & 12076 \\ 16904 & 2415 \end{array}\right),\left(\begin{array}{rr} 6442 & 1 \\ 6439 & 0 \end{array}\right),\left(\begin{array}{rr} 19317 & 4 \\ 19316 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 11594 & 1 \\ 3863 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 6721 & 4 \\ 13442 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[19320])$ is a degree-$1588427552194560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19320\Z)$.
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 | $2$ | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| $3$ | additive | $8$ | \( 2576 = 2^{4} \cdot 7 \cdot 23 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 23184 = 2^{4} \cdot 3^{2} \cdot 7 \cdot 23 \) |
| $7$ | split multiplicative | $8$ | \( 16560 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 115920dl
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 4830v1, its twist by $12$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.14219520.1 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | nonsplit | split | ord | ord | ord | ord | nonsplit | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | - | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.