Properties

Label 30.a2
Conductor $30$
Discriminant $5859375000$
j-invariant \( \frac{10316097499609}{5859375000} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -454, -544])
 
gp: E = ellinit([1, 0, 1, -454, -544])
 
magma: E := EllipticCurve([1, 0, 1, -454, -544]);
 

Minimal equation

Minimal equation

Simplified equation

\(y^2+xy+y=x^3-454x-544\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-454xz^2-544z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-587763x-23605938\) Copy content Toggle raw display (homogenize, minimize)

Mordell-Weil group structure

\(\Z/{2}\Z\)

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(\frac{87}{4}, -\frac{91}{8}\right) \) Copy content Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 30 \)  =  $2 \cdot 3 \cdot 5$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $5859375000 $  =  $2^{3} \cdot 3 \cdot 5^{12} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{10316097499609}{5859375000} \)  =  $2^{-3} \cdot 3^{-1} \cdot 5^{-12} \cdot 11^{3} \cdot 1979^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.56416598521635168028924232156\dots$
Stable Faltings height: $0.56416598521635168028924232156\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $1.1173160864138321498160760446\dots$
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: $ 2 $  = $ 1\cdot1\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Special value: $ L(E,1) $ ≈ $ 0.55865804320691607490803802232 $

Modular invariants

Modular form   30.2.a.a

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 2 q^{13} + 4 q^{14} - q^{15} + q^{16} + 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

For more coefficients, see the Downloads section to the right.

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 24
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is semistable. There are 3 primes of bad reduction:

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$3$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$5$ $2$ $I_{12}$ Non-split multiplicative 1 1 12 12

Galois representations

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

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 8.12.0.8
$3$ 3B.1.2 3.8.0.2

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ 2 3 5
Reduction type nonsplit split nonsplit
$\lambda$-invariant(s) 0 1 0
$\mu$-invariant(s) 0 1 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3, 4, 6 and 12.
Its isogeny class 30.a consists of 8 curves linked by isogenies of degrees dividing 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
$2$ \(\Q(\sqrt{6}) \) \(\Z/2\Z \oplus \Z/2\Z\) 2.2.24.1-150.1-e5
$2$ \(\Q(\sqrt{2}) \) \(\Z/4\Z\) 2.2.8.1-450.1-a3
$2$ \(\Q(\sqrt{3}) \) \(\Z/4\Z\) 2.2.12.1-150.1-a3
$2$ \(\Q(\sqrt{-3}) \) \(\Z/6\Z\) 2.0.3.1-300.1-a3
$3$ 3.1.243.1 \(\Z/6\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{3})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{-3})\) \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Z/12\Z\) Not in database
$4$ \(\Q(\zeta_{12})\) \(\Z/12\Z\) Not in database
$6$ 6.0.177147.2 \(\Z/3\Z \oplus \Z/6\Z\) Not in database
$6$ 6.2.90699264.1 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$6$ 6.2.30233088.6 \(\Z/12\Z\) Not in database
$6$ 6.2.11337408.2 \(\Z/12\Z\) Not in database
$8$ 8.0.3057647616.9 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$8$ 8.8.23592960000.1 \(\Z/8\Z\) Not in database
$8$ 8.0.1866240000.6 \(\Z/8\Z\) Not in database
$8$ \(\Q(\zeta_{24})\) \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$12$ 12.0.8226356490141696.17 \(\Z/6\Z \oplus \Z/6\Z\) Not in database
$12$ 12.0.8226356490141696.32 \(\Z/3\Z \oplus \Z/12\Z\) Not in database
$12$ 12.0.128536820158464.4 \(\Z/3\Z \oplus \Z/12\Z\) Not in database
$12$ 12.4.526486815369068544.25 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$16$ 16.0.149587343098087735296.14 \(\Z/4\Z \oplus \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ 16.0.891610044825600000000.6 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$16$ 16.0.3482851737600000000.5 \(\Z/24\Z\) Not in database
$18$ 18.0.617673396283947000000000000.3 \(\Z/18\Z\) 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.