Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+11378475x+12108723125\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+11378475xz^2+12108723125z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+14746502925x+564723388572750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-3925/4, 3925/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 114950 \) | = | $2 \cdot 5^{2} \cdot 11^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-157573475106549390625000$ | = | $-1 \cdot 2^{3} \cdot 5^{9} \cdot 11^{8} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{5885721311824151}{5692551601000} \) | = | $2^{-3} \cdot 5^{-3} \cdot 7^{3} \cdot 11^{-2} \cdot 19^{-6} \cdot 25793^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1382458952679181731774745854$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1345793026516827138461231298$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9490054959708933$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.17971136061282$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.067291293296658302999682683162$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.53833034637326642399746146530 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.538330346 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.067291 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 0.538330346\end{aligned}$$
Modular invariants
Modular form 114950.2.a.bk
For more coefficients, see the Downloads section to the right.
| Modular degree: | 14929920 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $11$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $19$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 25080 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 5226 & 1057 \\ 17765 & 13586 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 25070 & 25077 \\ 10059 & 8 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 12513 & 25072 \end{array}\right),\left(\begin{array}{rr} 15959 & 25068 \\ 20514 & 25007 \end{array}\right),\left(\begin{array}{rr} 16721 & 12 \\ 4180 & 1 \end{array}\right),\left(\begin{array}{rr} 22441 & 12 \\ 9246 & 73 \end{array}\right),\left(\begin{array}{rr} 25069 & 12 \\ 25068 & 13 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 25030 & 25071 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[25080])$ is a degree-$599107829760000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/25080\Z)$.
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 | $4$ | \( 3025 = 5^{2} \cdot 11^{2} \) |
| $3$ | good | $2$ | \( 3025 = 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $18$ | \( 4598 = 2 \cdot 11^{2} \cdot 19 \) |
| $11$ | additive | $72$ | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 114950q
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 2090e4, its twist by $-55$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-10}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{165}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.6988960.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-10}, \sqrt{-66})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.14675772375.3 | \(\Z/6\Z\) | 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 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | 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 |
| $18$ | 18.6.72833638949918086406975568711633302761177125000000000000.1 | \(\Z/18\Z\) | 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 | 11 | 19 |
|---|---|---|---|---|---|
| Reduction type | nonsplit | ord | add | add | nonsplit |
| $\lambda$-invariant(s) | 5 | 0 | - | - | 0 |
| $\mu$-invariant(s) | 1 | 0 | - | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.